Graph theory

It's just dots and lines

An example

Consider the following graph:

The vertex set is \(\{A,B,C,D,E\}\)

The edge set is \(\{AB, AE, BC, CE, ED\}\)

An example

An extremely useful way to represent a graph is called the adjacency matrix. This is the matrix with rows and columns labeled by the vertices (in the same order). The matrix has a \(1\) if the row and column indices are adjacent, and zero otherwise.

An example

An example

An example

A Disconnected example

\(x_{0} = \begin{bmatrix}1\\0\\0\\0\\0\end{bmatrix}\)     or    \(x_{0} = \begin{bmatrix}0\\0\\0\\1\\0\end{bmatrix}\)

Consider the previous algorithm starting with:

Is it Connected?

The graph has 16 edges, so we could delete each edge, then test again...

Is it Connected?

The Graph Laplacian

For a graph with vertices \(v_{1},v_{2},v_{3},\ldots,v_{n}\) the adjacency matrix \(A\) is given by

\[A_{i,j} = \begin{cases} 1 & v_i\text{ is adjacent to }v_{j},\\ 0 & \text{otherwise}.\end{cases}\]

The degree of vertex \(v_{i}\), denoted \(d(v_{i})\) is the number of edges incident with \(v_{i}\), that is \[d(v_{i}) = \sum_{i=1}^{n}A_{i,j}.\]

The Laplacian is the \(n\times n\) matrix \(L\) given by

\[L_{i,j} = \begin{cases} d(v_{i}) & i=j,\\ -A_{i,j} & i\neq j.\end{cases} \]

Consider the previous example:

For any vector \(x\in\mathbb{R}^{n}\) we have

\[x^{\top}Lx = [x_{1}\ \ x_{2}\ \ x_{3}\ \ \cdots\ \  x_{n}]\left[\begin{array}{ccccc}L_{1,1} & L_{1,2} & L_{1,3} & \cdots & L_{1,n}\\ L_{2,1} & L_{2,2} & L_{2,3} & \cdots & L_{2,n}\\[-1ex] L_{3,1} & L_{3,2} & L_{3,3} & & \vdots\\ \vdots & \vdots & & \ddots & \vdots\\ L_{n,1} & L_{n,2} & \cdots & \cdots & L_{n,n}\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ \vdots \\ x_{n}\end{array}\right]\]

\[= \sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}L_{i,j} = \sum_{i=1}^{n}x_{i}^{2}L_{i,i} - \sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}A_{i,j}\]

\[\left[\begin{array}{ccccc}d(v_{1}) & -A_{1,2} & -A_{1,3} & \cdots & -A_{1,n}\\ -A_{2,1} & d(v_{2}) & -A_{2,3} & \cdots & -A_{2,n}\\ -A_{3,1} & -A_{3,2} & d(v_{3}) & & \vdots\\ \vdots & \vdots & & \ddots & \vdots\\ -A_{n,1} & -A_{n,2} & \cdots & \cdots & d(v_{n})\end{array}\right]\]

\[= \sum_{i=1}^{n}x_{i}^{2}\sum_{j=1}^{n}A_{i,j} - \sum_{i=1}^{n}\sum_{i=1}^{n}x_{i}x_{j}A_{i,j} = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-x_{i}x_{j})A_{i,j}\]

Thus, since \(A_{i,j} = A_{j,i}\) we have

\[x^{\top}Lx = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-x_{i}x_{j})A_{i,j} = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{j}^{2}-x_{i}x_{j})A_{i,j}\]

Hence

\[x^{\top}Lx = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2})A_{i,j} = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}-x_{j})^{2}A_{i,j}.\]

For which vectors \(x\) is it the case that

\[x^{\top}Lx = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}-x_{j})^{2}A_{i,j} = 0?\]

Example

For this vector \(Lx = 0\):

For a connected graph:

Instead of \(Lx=0\), we look for \(x\neq 0\) such that \(Lx=\lambda x\) for the smallest \(\lambda\).

w/ \(\lambda = 0.3033351\)

Part 1:

Fundamental Concepts

Graphs

Definition 1.1.2. A graph \(G\) is a triple consisting of a vertex set \(V(G)\), an edge set \(E(G)\), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints.

Example. Note that the vertex set is \(\{v_{1},v_{2},v_{3},v_{4},v_{5}\}\) and the edge set is \(\{e_{1},e_{2},e_{3},e_{4},e_{5}\}\).

The endpoints of \(e_{4}\) are \(v_{3}\) and \(v_{4}\).

Example. The vertex set is \(\{v_{1},v_{2},v_{3},v_{4},v_{5}\}\) and the edge set is \(\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}\).

Both \(e_{1}\) and \(e_{6}\) have endpoints \(v_{1}\) and \(v_{2}\), thus they are multiple edges.

Simple Graphs

A graph without multiple edges and without loops is called a simple graph.

In a simple graph we can name the edges by their endpoints:

In the following graph, the edge set is \(\{v_{1}v_{2},v_{1}v_{3},v_{2}v_{4}, v_{3}v_{4}, v_{4}v_{5}\}\)

Definition 1.1.8. 

• The complement \(\overline{G}\) of a simple graph \(G\) is the simple graph with vertex set \(V(G)\) defined by \(uv\in E(\overline{G})\) if and only if \(uv\notin E(G)\).
• A clique in a graph is a set of pariwise adjacent vertices.
• An independent set (or stable set) in a graph is a set of pairwise nonadjacent vertices.

Example. 

\(G\)

\(\overline{G}\)

Example 1.1.7. Consider a group of 5 people. Is there always a group of three who are all mutual acquaintances or mutual strangers? In other words, for every simple graph on 5 vertices \(G\), does either \(G\) or \(\overline{G}\) contain a clique of size 3?

Definition 1.1.10. A graph \(G\) is bipartite if \(V(G)\) is the union of two disjoint (possibly empty) independent sets called partite sets of \(G\).

Example. 

Definition 1.1.15.

• A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if  they are consecutive in the list.
• A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along a circle.

Definition 1.1.16.

• A subgraph of a graph \(G\) is a graph \(H\) such that \(V(H)\subseteq V(G)\) and \(E(H)\subseteq E(G)\) and the assignment of endpoints in \(H\) is the same as in \(G\). We write \(H\subseteq G\) and say "\(G\) contains \(H\)".
• A graph \(G\) is connected if each pair of vertices in \(G\) belongs to a path contained in \(G\).

Example. 

Matrices and IsoMorphism

Consider the simple graph with

\[V(G) = \{a,b,c,d,e\}\quad\text{and}\quad E(G) = \{ab,bc,cd,ac,ae\}\]

Definition 1.1.17. Let \(G\) be a loopless graph with vertex set \(V(G) = \{v_{1},\ldots,v_{n}\}\) and edge set \(E(G) = \{e_{1},\ldots,e_{m}\}\). The adjacency matrix of \(G\), written \(A(G)\) is the \(n\times n\) matrix with entry in row \(i\), column \(j\) is the number of edges in \(G\) with endpoints \(\{v_{i},v_{j}\}\). The incidence matrix \(M(G)\) is the \(n\times m\) matrix in which the entry in row \(i\), column \(j\) is \(1\) if \(v_{i}\) is an endpoint of edge \(e_{j}\), and zero otherwise.

If vertex \(v\) is an endpoint of edge \(e\), then \(v\) and \(e\) are incident. The degree of a vertex (in a loopless graph) is the number of incident edges.

Example. Consider the simple graph with

\[V(G) = \{a,b,c,d,e\}\quad\text{and}\quad E(G) = \{ab,bc,cd,ac,ae\}\]

\[A(G) = \left[\begin{array}{ccccc} 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0\end{array}\right]\]

\[M(G) = \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 1 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\end{array}\right]\]

\[A(G_{1}) = \left[ \begin{array}{cccccc} 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right] \]

\[A(G_{2}) = \left[ \begin{array}{cccccc} 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right]\]

Are these graphs "the same"?

Graph isomorphism

Definition 1.1.20. An isomorphism from a simple graph \(G\) to a simple graph \(H\) is a bijection \(f:V(G)\to V(H)\) such that \(uv\in E(G)\) if and only if \(f(u)f(v)\in E(H)\). We say that "\(G\) is isomorphic to \(H\)", written \(G\cong H\), if there is an isomorphism from \(G\) to \(H\).

Consider the simple graph \(G\) with \[V(G) = \{a,b,c,d,e\}\] and \[E(G) = \{ab,bc,cd,ac,ae\}\]

Consider the simple graph \(H\) with \[V(H) = \{a,b,c,d,e\}\] and \[E(H) = \{bc,cd,de,bd,ba\}.\]

Proposition. If \(f\) is an isomorphism from \(G\) and \(H\), then for any \(v\in V(G)\) the degrees of \(v\) and \(f(v)\) are the same. Moreover, if

\[d_{1},d_{2},\ldots,d_{n}\quad\text{and}\quad d_{1}',d_{2}',\ldots,d_{n}'\]

are the lists of the degrees of their vertices of \(G\) and \(H\), respectively, listed in nonincreasing order, then these lists are identical.

Proof. Consider a vertex \(v\in V(G)\) with degree \(d\). This means that there are \(d\) other vertices \(v_{1},v_{2},\ldots, v_{d}\) that are adjacent to \(v\), that is \(vv_{i}\in E(G)\) for each \(i\in\{1,2,\ldots,d\}\). By the definition of an isomorphism, the vertices adjacent to \(f(v)\) are exactly \(f(v_{1}),\ldots,f(v_{d})\). Thus \(f(v)\) has degree \(d\). \(\Box\)

Example. None of the following graphs are isomorphic:

Definition. A permutation matrix is a square matrix with entries in \(\{0,1\}\) and exactly one \(1\) in each row and each column.

Example. Some permutation matrices:

\[\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}\]

\[\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\]

\[\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix}\]

\[\begin{bmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 &  1 & 0\end{bmatrix}\]

Multiplying a matrix by a permutation on the left permutes the rows:

\[\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} = \begin{bmatrix}  d & e & f\\ g & h & i\\ a & b & c\end{bmatrix}\]

Multiplying a matrix by a permutation on the right permutes the columns:

\[\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix} = \begin{bmatrix}  c & a & b\\  f & d & e\\ i & g & h\end{bmatrix}\]

Permutation matrices

Rows: \((1,2,3)\mapsto (3,1,2)\)

Columns: \((1,2,3)\mapsto (2,3,1)\)

\[\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}= \begin{bmatrix} i & g & h\\ c & a & b\\  f & d & e\end{bmatrix}\]

\(1\)

\(2\)

\(3\)

\(1\)

\(2\)

\(3\)

If we multiply by \(P\) on the right and \(P^{\top}\) on the left, then we permute the rows and columns in the same way:

Isomorphism: \((1,2,3)\mapsto (2,3,1)\)

\[\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix} 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 0\end{bmatrix} \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}= \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}\]

Consider permuting the rows and columns of an adjacency matrix:

Theorem. Two simple graphs \(G\) and \(H\) are isomorphic if and only if there is a permutation matrix \(P\) such that

\[A(H) = P^{\top}A(G)P.\]

Thus, if we can find a permutation matrix such that \(A(H) = P^{\top}A(G)P\), we can conclude \(G\) and \(H\) are isomorphic. If \(G\) and \(H\) are not isomorphic, it is harder to prove.

Definition 1.1.25. An isomorphism class of graphs is an equivalence class of graphs under the isomorphism relation.

Example. Every simple graph on \(2\) vertices is isomorphic to one of the following:

So, there are two isomorphism classes of graphs on \(2\) vertices.

Example. How many isomorphism classes are there on \(3\) vertices?

Connection

in

Graphs

Definition 1.2.2.

• A walk is a list \(v_{0},e_{1},v_{1},\ldots,e_{k},v_{k}\) of vertices and edges such that, for      \(i\in\{1,2,\ldots,k\}\), the edge \(e_{i}\) has endpoints \(v_{i-1}\) and \(v_{i}\).
• A trail is a walk with no repeated edge.
• A \(u,v\)-walk or \(u,v\)-trail has first vertex \(u\) and last vertex \(v\); the vertices \(u\) and \(v\) are called its endpoints.
• A \(u,v\)-path is a path whose endpoints are \(u\) and \(v\); the other vertices are called internal vertices.
• The length of a walk, trail, path, or cycle is its number of edges. A walk or trail is closed if its endpoints are the same.

Lemma 1.2.5. Every \(u,v\)-walk contains a \(u,v\)-path.

Proof. (On the board)

Example. A walk with a repeated vertex:

Definition 1.2.6. A graph is connected if for every pair of vertices \(u\) and \(v\), there is a \(u,v\)-path contained in the graph. Otherwise, we say that the graph is disconnected.

If \(G\) has a \(u,v\)-path, then we say that \(u\) is connected to \(v\) in \(G\). 

Definition 1.2.8. The components of a graph \(G\) are its maximal connected subgraphs. A component is trivial if it has no edges otherwise it is nontrivial. An isolated vertex is a vertex of degree \(0\).

Proposition 1.2.11. Every graph with \(n\) vertices and \(k\) edges has at least \(n-k\) components.

Note, if a graph \(G\) contains a \(u,v\)-path and a \(v,w\)-path, it is not clear that \(G\) contains a \(u,w\)-path. However, if we concatenate these paths, then we get a \(u,w\)-walk which contains a \(u,w\)-path by Lemma 1.2.5. Thus, the connection relation on vertices is transitive. It is also reflexive (vertices are connected to themselves), and symmetric (since paths are reversible).

Proof. Adding an edge to a graph decreases the number of connected components by at most \(1\). \(\Box\)

Definition 1.2.12.

• A cut-edge or cut-vertex of a graph is an edge or vertex, respectively, whose deletion increases the number of components.
• We write \(G-e\) or \(G-M\) for the subgraph of \(G\) obtained by deleting an edge \(e\) or a set of edges \(M\).
• We write \(G-v\) or \(G-S\) for the subgraph obtained by deleting a vertex \(v\) or a set of vertices \(S\).
• An induced subgraph is a subgraph obtained by deleting a set of vertices. 
• We write \(G[T]\) for \(G-\overline{T}\), where \(\overline{T} = V(G)-T\); this is called the subgraph of \(G\) induced by \(T\).

Theorem 1.2.14. An edge is a cut-edge if and only if it belongs to no cycle.

Proof. (On the board)

Note that this theorem is the statemet that the following two implications about an edge \(e\) in a graph \(G\) both hold:

\(e\) is a cut-edge \(\Rightarrow\) \(e\) belongs to no cycle

and

\(e\) belongs to no cycle \(\Rightarrow\) \(e\) is a cut-edge

Given an implication \(A\Rightarrow B\), it is often useful to consider the equivalent statement \(\sim B\Rightarrow \sim A\), which is called the contrapositive. (Here \(\sim A\) is the statement "\(A\) does not hold".)

Thus, we could instead prove the contrapositives of the above implications

\(e\) belongs to a cycle \(\Rightarrow\) \(e\) is not a cut-edge

and

\(e\) is a not a cut-edge \(\Rightarrow\) \(e\) belongs to a cycle

Eulerian Circuits

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Consider a path \(P\) in a graph \(G\). If there is another path \(Q\neq P\) such that \[P\subseteq Q\subseteq G,\] then we call \(Q\) an extension of \(P\). Since \(G\) only has finitely many vertices, we cannot extend a path forever. If a path \(P\subseteq G\) has no extension, then we call \(P\) a maximal path.

Proposition 1.2.27. Every even graph decomposes into cycles.

Note: This "decomposition" is a partition of the edges. The cycles may share vertices. Indeed, every vertex of degree \(\geq 2\) must be in more than one cycle.

Vertex Degrees  and counting

Definition 1.3.1.

• The degree  of a vertex \(v\) in a graph \(G\), written \(d_{G}(v)\) or \(d(v)\), is the number of edges incident to \(v\), except that each loop at \(v\) counts twice.
• The maximum degree is denoted \(\Delta(G)\), the minimum degree is denoted \(\delta(G)\), and \(G\) is regular if \(\Delta(G) = \delta(G)\).
• The graph \(G\) is \(k\)-regular if \(\Delta(G) = \delta(G) = k\).
• The neighborhood of \(v\), denoted \(N_{G}(v)\) or \(N(v)\), is the set of vertices adjacent to \(v\).

Regular graphs and Degree

\(1\)

\(1\)

\(2\)

\(2\)

\(2\)

\(4\)

\(\Delta(G) = 4\) and \(\delta(G) = 1\)

Example.

The Petersen Graph

Definition 1.1.36. The Petersen graph is the simple graph with vertices consisting of all \(2\)-element subsets of \(\{1,2,3,4,5\}\), that is,  

\[V(G) = \big\{\{1,2\},\{1,3\},\{1,4\},\{1,5\}, \{2,3\}, \{2,4\}, \{2,5\}, \{3,4\},\{3,5\},\{4,5\}\big\},\]

and two vertices are adjacent if the sets are disjoint.

Note that

\(\Delta(G) = \delta(G) = 3\)

Thus, the Petersen graph is \(3\)-regular.

Definition 1.1.27.

• The path and cycle with \(n\) vertices are denoted \(P_{n}\) and \(C_{n}\), respectively. An \(n\)-cycle is a cycle with \(n\) vertices.
• A complete graph is a simple graph whose vertices are pairwise adjacent. The complete graph on \(n\) vertices is denoted \(K_{n}\)
• A complete bipartite graph or biclique is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. When the partite sets have sizes \(r\) and \(s\), the graph is denoted \(K_{r,s}\)

Some important graphs

Definition 1.3.2.

• The order of a graph \(G\), written \(n(G)\), is the number of vertices in \(G\). 
• An \(n\)-vertex graph is a graph of order \(n\).
• The size of a graph \(G\), written \(e(G)\), is the number of edges in \(G\). 
• For \(n\in\mathbb{N}\), we define \([n]: = \{1,2,3,\ldots,n\}\).

Proposition 1.3.3. (Degree-Sum Formula) If \(G\) is a graph, then

\[\sum_{v\in V(G)}d(v) = 2e(G).\]

Proposition 1.3.9. If \(k>0\), then a  \(k\)-regular bipartite graph has the same number of vertices in each partite set.

regular bipartite graphs

Example. Consider the \(2\)-regular bipartite graph below

6-cycles in the petersen graph

6-cycles in the petersen graph

\(\#\) cycles \(\geq \#\) vertices:

6-cycles in the petersen graph

\(\#\) cycles \(\leq \#\) vertices:

Proposition 1.3.11 For a simple graph \(G\) with vertices \(v_{1},\ldots,v_{n}\) and \(n\geq 3\),

\[e(G) = \frac{1}{n-2}\sum_{i=1}^{n}e(G-v_{i})\quad\text{and}\quad d_{G}(v_{j}) = \frac{1}{n-2}\sum_{i=1}^{n} e(G-v_{i}) - e(G-v_{j})\]

A subgraph obtained by deleting a single vertex is called a vertex-deleted subgraph.

Vertex-deleted subgraphs

Example. Consider the graph \(G\) and its vertex-deleted subgraphs:

Extremal Problems

Proposition 1.3.13. The minimum number of edges in a connected graph with \(n\) vertices is \(n-1\).

degree sequences

Definition 1.3.27. The degree sequence of a graph is the list of vertex degrees, usually written in nonincreasing order \(d_{1}\geq \cdots\geq d_{n}\). 

Definition 1.3.29. A graphic sequence is a list of nonnegative number that is the degree sequence of some simple graph.

Graphic sequences

What about degree sequences of simple graphs?

Example. The degree sequences of both graphs are \(3,2,2,1,1,1\)

\[\begin{array}{r|r|r|r|r|r|r|r|r}i & 1 & 2 & 3 & & \Delta+1 & \Delta+2 & & n\\\hline\\ d & \phantom{\Delta = }d_{1} & d_{2} & d_{3} & \ldots &  d_{\Delta+1} & d_{\Delta+2} & \ldots, & d_{n}\\\\\hline \\ d' & & \phantom{d_{2}-1} & \phantom{d_{3}-1} & \phantom{\ldots} &  \phantom{d_{\Delta+1}-1} & \phantom{d_{\Delta+2}} & \phantom{\ldots} & \phantom{d_{n}} \end{array}\]

How to build \(d'\) from \(d\): Suppose \(d_{1}\geq \cdots\geq d_{n}\)

The sequence \(d = (3,2,2,2,1)\) is graphic since:

Deleting the degree 3 \(\rightarrow\)

So \((2,1,1,0)\) is graphic, but why is \(d' = (1,1,1,1)\) graphic?

Why does \(d\) graphic \(\Rightarrow\) \(d'\) graphic?

\[\begin{array}{c|ccccc} \text{vertices} & v_{1} & v_{2} & v_{3} & v_{4} & v_{5}\\\hline  \text{degrees} & 3 & 2 & 2 & 2 & 1\end{array}\]

\(N(v_{1}) = \{v_{2},v_{4},v_{5}\}\)

Directed graphs

Definition 1.4.2. 

A directed graph or digraph \(G\) is a triple consisting of a vertex set \(V(G)\), an edge set \(E(G)\), and  function assigning each edge an ordered pair of vertices. The first vertex in the ordered pair is the tail of the edge, and the second is the head; together, they are the endpoints. We say that an edge is from its tail to its head.

What is a Directed graph

Example. Let \(V(G) = \{v_{1},v_{2},v_{3},v_{4},v_{5}\}\) and \(E(G) = \{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}\)

and the function that assigns edges to points is given by

\[f(e_{1}) = (v_{1},v_{2}),\quad f(e_{2}) = (v_{2},v_{3}),\quad f(e_{3}) = (v_{3},v_{5})\]

\[f(e_{4}) = (v_{4},v_{4}),\quad f(e_{5}) = (v_{2},v_{1}),\quad f(e_{6}) = (v_{3},v_{5})\]

Definition 1.4.3. In a digraph:

• A loop is an edge whose endpoints are equal.
• Multiple edges are edges having the same ordered pair of endpoints.
• The digraph is simple if each ordered pair is the head and tail of at most one edge (Note: loops are allowed!)

Definition 1.4.6. A digraph \(G\) is a path if it is simple, and all of the vertices can be listed (without repetition) \[v_{1},v_{2},\ldots,v_{n}\] such that \[E(G) = \{v_{1}v_{2}\ ,\ v_{2}v_{3}\ ,\ v_{3}v_{4}\ ,\ \ldots\ ,\ v_{n-1}v_{n}\}\]

A digraph is a cycle if it is simple, and all of the vertices can be listed (without repetition) \[v_{1},v_{2},\ldots,v_{n}\] such that \[E(G) = \{v_{1}v_{2}\ ,\ v_{2}v_{3}\ ,\ v_{3}v_{4}\ ,\ \ldots\ ,\ v_{n-1}v_{n}\ ,\ v_{n}v_{1}\}\]

Cycle

Path

Neither

Example: Markov Chain

\[P=\left[\begin{array}{rrr}0 & 0.9 & 0.1\\ 1 & 0 & 0\\ 0.1 & 0.1 & 0.8 \end{array}\right]\]

Given a vector of nonnegative entries \[\mathbf{x} = [x_{A}\ \ x_{B}\ \ x_{C}]\quad\text{ such that }\quad x_{A}+x_{B}+x_{C}=1,\] we interpret \(x_{i}\) as the probability that we're in state \(i\)

Then \(\mathbf{x}P\) gives the probabilities that we're in each state after another step.

Example: Markov Chain

\[P=\left[\begin{array}{rrr}0 & 0.9 & 0.1\\ 1 & 0 & 0\\ 0.1 & 0.1 & 0.8 \end{array}\right]\]

Let \(\mathbf{x}_{n}\) be the vector of probabilities after \(n\) steps.

If we start at \(A\), then \(\mathbf{x}_{0} = [1\ \ 0\ \ 0]\), and

\[\mathbf{x}_{1} = \mathbf{x}_{0}P = [0\ \ 0.9\ \ 0.1]\]

Digraph degrees

Definition 1.4.17. Let \(v\) be a vertex in a digraph.

• The outdegree \(d^{+}(v)\) is the number of edges with tail \(v\).
• The indegree is the number of edges with head \(v\). 
• The out-neighborhood or successor set \(N^{+}(v)\) is the set \[\{x\in V(G) : v\to x\}.\]
• The in-neighborhood or predecessor set \(N^{-}(v)\) is the set \[\{x\in V(G) : x\to v\}.\]
• The minimum and maximum indegree are \(\delta^{-}(G)\) and \(\Delta^{-}(G)\); for outdegree we use \(\delta^{+}(G)\) and \(\Delta^{+}(G)\).

Eulerian Digraphs

Definition 1.4.22.

• An Eulerian trail in a digraph is a trail containing all edges.
• An Eulerian circuit is a closed trial containing all edges.
• A digraph is Eulerian if it has an Eulerian circuit.

We have already seen that graphs are Eulerian if and only if they are even. What about digraphs?

Trees

Definition 2.1.1.

• A graph with no cycle is acyclic.
• A forest is an acyclic graph.
• A tree is a connected acyclic graph.
• A leaf is a vertex of degree \(1\).
• A spanning subgraph of \(G\) is a subgraph with vertex set \(V(G)\).
• A spanning tree is a spanning subgraph that is also a tree.

Some floral definitions

\[\underbrace{\hspace*{1.6in}}\]

\[\underbrace{\hspace*{1.6in}}\]

\[\underbrace{\hspace*{1.2in}}\]

Forest

Tree

Lemma 2.1.3. Every tree with at least two vertices has at least two leaves. Deleting a leaf from an \(n\)-vertex tree produces a tree with \(n-1\) vertices.

Theorem 2.1.4. For an \(n\)-vertex graph \(G\) (with \(n\geq 1\)), the following are equivalent:

A) \(G\) is connected and has no cycles, that is, \(G\) is a tree.

B) \(G\) is connected and has \(n-1\) edges.

C) \(G\) has \(n-1\) edges and no cycles.

D) For \(u,v\in V(G)\), \(G\) has exactly one \(u,v\)-path.

The next theorem give several alternate characterizations of graphs that are trees. Note that each of A, B, and C is the statement that two of the following hold: 1) \(G\) is acyclic, 2) \(G\) is connected, 3) \(G\) has \(n-1\) edges.

Some other characterizations of trees

Corollary 2.1.5.

     a) Every edge of a tree is a cut edge.

     b) Adding one edge to a tree forms exactly one cycle.

     c) Every connected graph contains a spanning tree.

Proposition 2.1.6. If \(T\) and \(T'\) are spanning trees of a connected graph \(G\), and \(e\in E(T)-E(T')\), then there is an edge \(e'\in E(T')-E(T)\) such that \(T-e+e'\) is a spanning tree of \(G\).

Proposition 2.1.7. If \(T\) and \(T'\) are spanning trees of a connected graph \(G\), and \(e\in E(T)-E(T')\), then there is an edge \(e'\in E(T')-E(T)\) such that \(T'+e-e'\) is a spanning tree of \(G\).

Spanning Trees

Suppose \(T\) is a tree and \(G\) is a simple graph. What must be true about \(T\) and \(G\) to ensure that \(T\) is a subgraph of \(G\).

If \(T\) has one vertex adjacent to all of the others, then \(T\) is called a star.

Clearly, \(T\) is a subgraph of \(G\) if and only if \(G\) has a vertex of degree \(e(T)\), that is, \(\Delta(G)\geq e(T)\).

Is is sufficient? No! \(P_{4}\) is not s subgraph of the star with \(4\) vertices.

What about \(\delta(G)\geq e(T)\)?

\(G\)

Not a subgraph of \(G\)

 Subgraph of \(G\)

\(G\)

What about the other trees with 4 edges?

Proposition 2.1.8. If \(T\) is a tree with \(k\) edges, and \(G\) is a simple graph with \(\delta(G)\geq k\), then \(T\) is a subgraph of \(G\)

\(T\)

Proposition 2.1.8. If \(T\) is a tree with \(k\) edges, and \(G\) is a simple graph with \(\delta(G)\geq k\), then \(T\) is a subgraph of \(G\)

Distance in Graphs

Definition 2.1.9.

• If \(G\) has a \(u,v\)-path, then the distance from \(u\) to \(v\), written \(d_{G}(u,v)\) or simply \(d(u,v)\), is the length of the shortest \(u,v\)-path.
• If there is no \(u,v\)-path in \(G\), then \(d(u,v) = \infty\).
• The diameter of \(G\) is given by \[\operatorname{diam}G = \max\{d(u,v) : u,v\in V(G)\}.\]
• The eccentricity of a vertex \(u\) is given by \[\epsilon(u) = \max\{d(u,v) : v\in V(G)\}.\]
• The radius of a graph \(G\) is given by \[\operatorname{rad}G = \min\{\epsilon(u) : u\in V(G)\}.\]

\(b\)

\(a\)

\(c\)

\(e\)

\(f\)

\(g\)

\(d(a,b)=1\)

\(d(a,c)=2\)

\(d(a,e)=3\)

\(d(a,f)=2\)

\(d(a,g)=1\)

\(d(a,a)=0\)

\(\operatorname{diam}G = 3\)

\(\epsilon(a) = 3\)

\(\operatorname{rad}G = 3\)

Example.

Theorem 2.1.11. Suppose \(G\) is a simple graph. If \(\operatorname{diam}G\geq 3\), then \(\operatorname{diam}\overline{G}\leq 3\).

Proof.  Since \(\operatorname{diam}G>2\), there are two nonadjacent vertices \(u\) and \(v\) with no common neighbor.

Hence, for \(x\in V(G)-\{u,v\}\) either \(x\not\leftrightarrow u\) or \(x\not\leftrightarrow v\).

Note that \(u\) and \(v\) are adjacent in \(\overline{G}\). And any other vertex \(x\) is adjacent to either \(u\) or \(v\) in \(\overline{G}\). 

Therefore, for any two vertices, there is a path of lenghth at most \(3\) in \(\overline{G}\). \(\Box\)

Definition 2.1.12. The center of a graph \(G\) is the subgraph induced by the vertices of minimum eccentricity.

Theorem 2.1.13. The center of a tree is either a vertex (\(K_{1}\)) or an edge (\(K_{2}\)).

the center

If we want a graph with all vertices close together, then we want a small diameter.

On the other hand, if we want a graph where most vertices are close together, then we might want the average distsnce between points to be small. That is, if \(G\) is a graph with \(n\) vertices, then we want

\[\frac{2}{n(n+1)}\sum_{v,u\in V(G)}d_{G}(v,u)\]

to be small. Equivalently, we want the Wiener index:

\[D(G): = \sum_{v,u\in V(G)}d_{G}(v,u)\]

to be small.

The next theorem gives us the extremes of this index among all trees.

Theorem 2.1.14. For trees with \(n\) vertices, the Wiener index is minimized by stars and maximized by paths, both uniquely. That is, if \(T\) is a tree with \(n\) vertices, then

\[ (n-1)^2\leq D(T)\leq {n+1\choose 3},\]

moreover, if \(D(T) = (n-1)^2\) then \(T\cong K_{1,n-1}\) and if \(D(T) = {n+1\choose 3}\) then \(T\cong P_{n}\).

Optimization and Trees

Weighted graphs

A weighted graph is a graph \(G\) together with a nonnegative number for each edge.

Kruskal's Algorithm

Suppose \(G\) is a connected, weighted graph with \(n\) vertices:

Theorem 2.3.3. In a connected weighted graph \(G\), Kruskal's Algorithm constructs a minimum-weight spanning tree.

Dijkstra's Algorithm

\(v\)

\(u\)

What is the shortest path from vertex \(v\) to vertex \(u\)?

Dijkstra's Algorithm

More generally, take the distance between vertices \(d(u,v)\) to be the smallest sum of weights among all \(u,v\)-paths.

What is \(d(u,v)\)?

\(v\)

\(u\)

Start with a list of distances \(0\) for \(v\) and \(\infty\) for the rest. 

1. Pick unvisited vertex w/ smallest distance (currently)
2. For all neighbors update distance to neighbors, if shorter

Theorem 2.3.7. Given a graph \(G\) and a vertex \(v\in V(G)\), Dijkstra's Algorithm computes \(d(v,z)\) for each vertex \(z\in V(G)\).

Dijkstra's Algorithm (It works!)

Dijkstra's Algorithm (It works!)

Proof continued.

• By induction \(t_{k-1}(z_{j})\) is the length of the shortest \(v,z_{j}\)-path through vertices \(z_{1},\ldots,z_{k-2}\).
• Since \(t_{k-1}(z_{k-1})\geq t_{k-1}(z_{j})\), the shortest \(v,z_{j}\)-path through \(z_{1},\ldots,z_{k-1}\) does not go through \(z_{k-1}\).
• Thus, \(t_{k-1}(z_{j})\) is the length of the shortest \(v,z_{j}\)-path through \(z_{1},\ldots,z_{k-1}\). 

Breadth-first search

\(v\)

\(u\)

For an unweighted graph, Dijkstra's algorithm is called a Breadth-First search.

However, we just take the next unvisited vertex that we've "seen" instead of the vertex with smallest tentative weight.

Matchings and Covers

Definition 3.1.1.

• A matching in a graph \(G\) is a set of non-loop edges with no shared endpoints.
• The vertices incident to the edges of a matching \(M\) are saturated by \(M\); the others are unsaturated.
• A perfect matching in a graph is a matching that saturates every vertex.

Example. 

Example. If a graph has a vertex with degree \(0\), then it contains no perfect matching.

Give an example of a graph \(G\) such that \(\delta(G)\geq 1\), but no perfect matching?

Example 3.1.2. A perfect matching in \(K_{n,n}\) corresponds to a bijection of the partite sets.

If \(X = \{x_{1},x_{2},\ldots,x_{n}\}\) and \(Y=\{y_{1},y_{2},\ldots,y_{n}\}\) are the partite sets of vertices, then a bijection \(f:X\to Y\) gives a perfect matching, namely, the edges \(xf(x)\) for each \(x\in X\).

Example 3.1.3. Why is there no perfect matching in \(K_{3}\), \(K_{5}\), \(K_{7},\ldots\)?

Definition 3.1.4. A maximal matching in a graph is a matching that cannot be enlarged by adding an edge. A maximum matching is a matching of maximum size among all matchings in the graph.

Maximal matching

(not maximum):

Maximum matching:

Definition 3.1.7. If \(G\) and \(H\) are graphs with a common vertex set \(V\), then the symmetric difference \(G\triangle H\) is the graph with vertex set \(V\) whose edges are all those appearing in exactly one of \(G\) and \(H\). We similarly define the notion of symmetric difference for sets of edges.

Lemma 3.1.9. Every component of the symmetric difference of two matchings is a path or an even cycle.

\(\triangle\)

\(\triangle\)

\(=\)

\(=\)

Theorem 3.1.10. A matching \(M\) in a graph \(G\) is a maximum matching in \(G\) if and only if \(G\) has no \(M\)-augmenting path.

Saturating partite sets

Given a bipartite graph \(G\), and one of the partite sets \(X\), one might ask whether there is a matching that saturates \(X\).

Consider the following graph:

The blue edges are a matching that saturates \(X:=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}\).

If \(S\subset X\), then clearly the number of vertices adjacent to \(S\) must be \(\geq\) the number of vertices in \(S\).

Theorem 3.1.11. (Hall's Theorem) An \(X,Y\)-bigraph \(G\) has a matching that saturates \(X\) if and only if \(|N(S)|\geq |S|\) for all \(S\subseteq X\).

Recall:

• A bipartite graph \(G\) with partite sets \(X\) and \(Y\) is called an \(X,Y\)-bigraph.
• Given a set of vertices \(S\), the set neighborhood of \(S\) is the set \[N(S) = \{v\in V(G) : v\leftrightarrow w\text{ for some }w\in S\}.\]
• Given a finite set \(X\), the cardinality of \(S\) is the number of elements in \(S\), denoted \(|S|\).

Hall's Matching condition

Corollary (the Marriage Theorem). An \(X,Y\)-bigraph \(G\) contains a perfect matching if and only if both of the following hold:

• \(|X|=|Y|\)
• For every \(S\subseteq X\), \(|N(S)|\geq |S|\).

Does the following bigraph have a perfect matching?

Corollary 3.1.13. For \(k>0\), every \(k\)-regular bipartite graph has a perfect matching.

Does the following bigraph have a perfect matching?

Min-max Theorems

Definition 3.1.14. A vertex cover of a graph \(G\) is a set \(Q\subseteq V(G)\) that contains at least one endpoint of every edge. We say that the vertices in \(Q\) cover \(E(G)\).

Example. In each of the following graphs, the blue vertices comprise a vertex cover:

blue

Theorem 3.1.16. If \(G\) is a bipartite graph, then the maximum size of a matching in \(G\) equals the minimum size of a vertex cover of \(G\).

1. The blue edges form a maximum matching \(M\).
2. A vertex cover \(Q\) must include one endpoint of each of these edges.
3. There might be more vertices, not incident with any edge in \(M\).

Proof. Suppose \(X\) and \(Y\) are the partite sets in \(G\). Let \(Q\) be a vertex cover of minimum size.

• Set \(R = Q\cap X\quad\text{and}\quad T = Q\cap Y\).
• Let \(H\) and \(H'\) be the subgraphs induced by \(R \cup (Y-T)\) and \(T\cup (X-R)\), respectively.

Theorem 3.1.16. If \(G\) is a bipartite graph, then the maximum size of a matching in \(G\) equals the minimum size of a vertex cover of \(G\).

Proof. For each \(S\subseteq R\) we consider \(N_{H}(S)\).

Theorem 3.1.16. If \(G\) is a bipartite graph, then the maximum size of a matching in \(G\) equals the minimum size of a vertex cover of \(G\).

Independent Sets and Covers

Definition. The independence number of a graph is the maximum size of an independent set of vertices.

Is the independence number of a tree equal to the number of leaves?

Definition 3.1.19. An edge cover of a graph \(G\) is a set of edges such that every vertex of \(G\) is incident to some edge of \(L\).

How many edges are in the smallest edge cover of the following graph?

Lemma 3.1.21. In a graph \(G\), a subset \(S\subseteq V(G)\) is an independent set if and only if \(\overline{S}\) is a vertex cover, and hence \(\alpha(G)+\beta(G) = n(G)\).

Definition 3.1.20. 

• \(\alpha(G)\) is the maximum size of an independent set in \(G\)
• \(\alpha'(G)\) is the maximum size of an matching in \(G\)
• \(\beta(G)\) is the minimum size of an vertex cover in \(G\)
• \(\beta'(G)\) is the minimum size of an edge cover in \(G\)

Proof. We have the following equivalent statements:

• \(S\) is independent
• every edge has at most one endpoint in \(S\)
• every edge has at least one endpoint in \(\overline{S}\).   \(\Box\)

Theorem 3.1.22. If \(G\) is a graph without isolated vertices, then \[\alpha'(G) + \beta'(G) = n(G).\]

Definition 3.1.20. 

• \(\alpha'(G)\) is the maximum size of an matching in \(G\)
• \(\beta'(G)\) is the minimum size of an edge cover in \(G\)

Proof that \(\alpha'(G)+\beta'(G)\leq n(G)\).

• Let \(M\) be a maximum matching in \(G\). We will construct an edge cover of size \(n(G)-|M|\). 
• For each vertex \(v\) not saturated by \(M\), add to \(M\) an edge incident with \(v\). This produces an edge cover, call it \(\tilde{M}\)
• Each edge in \(M\) covers two vertices, the rest cover exactly one, thus \[|\tilde{M}| = |M|+(n(G)-2|M|) = n(G) - |M|.\]
• Thus \(\beta'(G)\leq |\tilde{M}|=n(G)-|M| =n(G)-\alpha'(G).\)

Theorem 3.1.22. If \(G\) is a graph without isolated vertices, then \[\alpha'(G) + \beta'(G) = n(G).\]

Definition 3.1.20. 

• \(\alpha'(G)\) is the maximum size of an matching in \(G\)
• \(\beta'(G)\) is the minimum size of an edge cover in \(G\)

Proof that \(\alpha'(G)+\beta'(G)\geq n(G)\).

• Suppose \(L\) is a minimum edge cover in \(G\).
• For an edge \(e\), if both of its endpoints are covered by edges in \(L\), then      \(e\notin L\), since otherwise \(L-\{e\}\) would be a smaller edge cover.
• Thus, the graph formed by the edges in \(L\) contains no paths of length \(>2\). It's a forest and each component is a star!
• For each component, take one edge, this forms a matching \(\tilde{L}\).
• If \(k\) is the number of components, then \(n(G) = |L|+k\) (each edge counts one leaf, but we also must count the central vertex in each star).
• Thus, \(\alpha'(G)\geq |\tilde{L}| = k = n(G)-|L| = n(G)-\beta'(G).\ \Box\)

Maximum Bipartite Matching

Augmenting Path Algorithm

\(\left\{\begin{array}{l}\text{\(G\) - an \(X,Y\)-bigraph}\\ \text{\(M\) - a matching in \(G\)}\\ \text{\(U=\) verts in \(X\) unsat. by \(M\)}\end{array}\right.\)

Inputs: 

Initialize: 

\(\left\{\begin{array}{l}S=U\\ \text{\(T=\varnothing\)}\\ K = \varnothing\end{array}\right.\)

Iteration: If \(K = S\) then by the next theorem \(T\cup (X-S)\) is a minimum cover, and \(M\) as a maximum matching.

If \(K\subsetneq S\), the select \(x\in S-K\). Go through each \(y\in N(x)\) such that \(xy\notin M\).

1.  If \(y\) is unsaturated by \(M\): there's an \(M\)-alt. path to from \(U\) to \(x\), adding the edge \(xy\) gives an \(M\)-augmenting path.
2. If \(y\) is sat. by \(M\), then \(\exists w\in X\) an \(M\)-alternating path from \(U\) to \(w\). Add \(w\) to \(S\) and \(y\) to \(T\).

Note: \(S\subseteq X\), \(T\subseteq Y\) and \(K\subseteq S\) is the "marked vertices".

Example.

Example.

Theorem 3.2.2. Repeatedly applying the Augmenting Path Algorithm to a bipartite graph produces a matching and a vertex cover of equal size.

Connectivity

Definition 4.1.1. 

• A separating set or vertex cut of a graph \(G\) is a set \(S\subseteq V(G)\) such that \(G-S\) has more than one component.
• The connectivity of \(G\), written \(\kappa(G)\), is the minimum size of a separating set, or \(n(G)-1\) if no separating set exists.
• A graph is \(k\)-connected if its connectivity is at least \(k\).

In this section all graphs have no loops.

Lemma. \(\kappa(G)\leq \delta(G)\)

Harary graphs

Definition (see Example 4.1.4.). Given \(k<n\) the graph \(H_{k,n}\) has \(n\) vertices placed equidistant around a circle, each vertex is connected to the \(\lfloor k/2\rfloor\) vertices closest to it around the circle.

If \(k\) is odd and \(n\) is even, add an edge between antipodal vertices as well.

If \(k\) is odd and \(n\) is odd, label the vertices \(\{0,1,2,\ldots,n\}\), and add an edge between vertices \(i\) and \(i+(n-1)/2\) for \(i=0,1,\ldots,(n-1)/2\).

The graphs \(H_{4,8}\), \(H_{5,8}\), and \(H_{5,9}\) are shown here:

Harary graphs

Theorem 4.1.5. \(\kappa(H_{k,n}) = k\), and hence the minimum number of edges in a \(k\)-connected graph on \(n\) vertices is \(\lceil kn/2\rceil\).

Definition 4.1.7.

• A disconnecting set of edges is a set \(F\subseteq E(G)\) such that \(G-F\) has more than one component.
• A graph is \(k\)-edge-connected if every disconnecting set has at least \(k\) edges.
• The edge-connectivity of \(G\), denoted \(\kappa'(G)\), is the is minimum size of a disconnecting set.
• Given \(S,T\subseteq V(G)\), we write \([S,T]\) for the set of edges having one endpoint in \(S\) and one endpoint in \(T\). 
• An edge cut is an edge set of the form \([S,\overline{S}]\), where \(S\neq\varnothing\) and \(S\subsetneq V(G)\).

Given disconnecting set \(F\subseteq E(G)\), we can delete one endpoint for each edge in \(F\) to disconnect the graph, right?

Disconnecting sets vs. Separating sets

Theorem 4.1.9. If \(G\) is a simple graph, then

\[\kappa(G)\leq \kappa'(G)\leq \delta(G).\]

Disconnecting sets vs. Separating sets

Proof.  The set of edges incident with a vertex of degree \(\delta(G)\) is an edge cut, thus \(\kappa'(G)\leq \delta(G)\).

• blue edges \(\subseteq [S,\overline{S}]\)
• \(\kappa'(G) = [S,\overline{S}]\geq |T|\geq \kappa(G)\)

Vertex Coloring

How many colors do we need so that every bordering territory is a different color?

How many colors do we need so that every bordering territory is a different color?

Four colors

Its a graph theory problem!

Its a graph theory problem!

Its a graph theory problem!

Definition 5.1.1.

• A \(k\)-coloring of a graph \(G\) is a labeling \(f: V(G)\to S\), where \(|S| = k\).
• The labels are colors; the vertices of one color form a color class.
• A \(k\)-coloring is proper if adjacent vertices have different labels.
• A graph is \(k\)-colorable if it has a proper \(k\)-coloring.
• The chromatic number \(\chi(G)\) is the least \(k\) such that \(G\) is \(k\)-colorable.

Proposition. A graph is \(2\)-colorable if and only if it is bipartite.

Proposition. \(C_{5}\) is not \(2\)-colorable.

Proof. Assume toward a contradiction there exists a proper \(2\)-coloring \(f~:~V(G)\to \{1,2\}\). Label the vertices \(\{0,1,2,3,4\}\) such that \(i\) and \(j\) are adjacent if and only if \(i-j\equiv \pm 1\) (mod \(5\)).  By the pigeonhole principle, at least \(3\) vertices are labeled the same color. Let these vertices be \(a,b,c\) where \(0\leq a<b<c\leq 4\). Note that \(b\geq a+2\) and \(c\geq b+2\). Therefore \(4\geq c\geq a+4\geq 4\). We conclude \(a=0\) and \(c=4\), and since \(0-4\equiv 1\) (mod \(5\)), we see \(a\leftrightarrow c\). Thus \(f\) is not a proper coloring, which is a contradiction. \(\Box\)

The US map has a chromatic number of \(4\)

Definition 5.1.6. The clique number of a graph \(G\), written \(\omega(G)\), is the maximum size of a set of pairwise adjacent vertices (a clique) in \(G\).

Theorem 5.1.7. For every graph \(G\),

\[\chi(G)\geq \omega(G) \quad\text{and}\quad\chi(G)\geq n(G)/\alpha(G).\]

Bounds on the chromatic number

Proof. A proper coloring of a clique on \(k\) vertices requires \(k\) colors. Thus, the first inequality is clear.

Each color class is an independent set. Let \(V_{1},\ldots,V_{\chi(G)}\) be the color classes of a minimum proper coloring, then \(|V_{i}|\leq \alpha(G)\), and \[\chi(G)\alpha(G)\geq |V_{1}|+|V_{2}|+\cdots+|V_{\chi(G)}| = n(G).\ \Box\]

Example. Consider the join of \(C_{5}\) and \(K_{3}\), which is denoted \(C_{5}\vee K_{3}\) (see Definition 3.3.6)

A better bound

Theorem 5.1.13. \(\chi(G)\leq \Delta(G)+1\).

Planar graphs

Example 6.1.1. Alice, Bob, and Chris each need to hook up gas, water, and electricity to their homes. The hookups don't need to be straight, but the lines cannot cross. Can it be done?

Proposition 6.1.2. The graphs \(K_{5}\) and \(K_{3,3}\) cannot be drawn in the plane without edges crossing.

Definition 6.1.4. 

• A graph is planar if it has a drawing without crossings.
• Such a drawing is a planar embedding of \(G\).
• A plane graph is a particular planar embedding of a planar graph.
• A curve is closed if its first and last points are the same.
• It is simple if it has no repeated points other than possibly the first and last.

Definition 5.2.19. A subdivision  of \(G\) is a graph obtained by successively replacing edges with paths. That is, an edge \(xy\in E(G)\) is replaced with the path \(x,z_{1},\ldots,z_{n},y\), where the new vertices \(z_{1},\ldots,z_{n}\) are only adjacent to the adjacent vertices in the path.

\(G\)

subdivision of \(G\)

Characterization of Planar graphs

Proposition 6.2.1. If a graph \(G\) has a subgraph that is a subdivision of \(K_{5}\) or \(K_{3,3}\), then it is nonplanar.

Theorem 6.2.2. A graph is planar if and only if it does not contain a subdivision of \(K_{5}\) or \(K_{3,3}\).

Proof. If a subdivision of a graph is planar, then removing degree \(2\) vertices and replacing them with edges would result in a planar graph. Thus, if a subdivision of \(K_{5}\) or \(K_{3,3}\) were planar, then \(K_{5}\) or \(K_{3,3}\) would be planar, which they are not. \(\Box\)

Faces of a planar graph and Euler's formula

Definition 6.1.5.

• An open set in the plane is a set \(U\subseteq\mathbb{R}^{2}\) such that for every \(p\in U\), there is a small distance \(\varepsilon\) such that all points with in \(\varepsilon\) from \(p\) are also in \(U\).
• A region is an open set that contains a continuous path between any two points in the set.
• The faces of a plane graph are the maximal regions of the plane that contain no point used in the embedding.

Theorem 6.1.21 (Euler's Formula). If a connected plane graph \(G\) has exactly \(n\) vertices, \(e\) edges, and \(f\) faces, then \(n-e+f=2\).

Faces of a planar graph and Euler's formula

Theorem 6.1.23. If \(G\) is a simple planar graph with at least \(3\) vertices, then \(e(G)\leq 3n(G)-6\). If \(G\) is triangle free, then \(e(G)\leq 2n(G)-4\).

An application of Euler's Formula

Theorem 6.1.23. If \(G\) is a simple planar graph with at least \(3\) vertices, then \(e(G)\leq 3n(G)-6\). If \(G\) is triangle free, then \(e(G)\leq 2n(G)-4\).

An application of Euler's Formula

Example. Note \[e(K_{5}) = 10>9=3\cdot 5-6 = 3n(K_{5})-6,\] which implies \(K_{5}\) is not planar.

Similarly \[e(K_{3,3})=9>8 = 2\cdot 6-4 = 2n(K_{3,3}) - 4,\] and since \(K_{3,3}\) is triangle free, \(K_{3,3}\) is not planar.

Theorem 6.1.23. If \(G\) is a simple planar graph with at least \(3\) vertices, then \(e(G)\leq 3n(G)-6\). If \(G\) is triangle free, then \(e(G)\leq 2n(G)-4\).

An application of Euler's Formula

Corollary. If \(G\) is simple and planar, then \(\delta(G)\leq 5\).

Proof. Assume toward a contradiction that \(\delta(G)>5\). For each \(v\in V(G)\) we have \(d(v)\geq 6\). By the degree-sum formula we have \[2e(G) \geq 6n(G).\]

By Theorem 6.1.23 \[3n(G)\leq e(G)\leq 3n(G)-6.\]

This is a contradiction, and shows \(\delta(G)\leq 5\). \(\Box\)

The Five color theorem

Theorem 6.3.1. Every planar graph is \(5\)-colorable.

Proof. We will use induction on \(n(G)\). For the basis step we note that any graph with \(n(G)\leq 5\) is \(5\)-colorable.

• In the \(5\)-coloring of \(G-v_{0}\) each of these vertices has a different color, so suppose \(v_{i}\) is colored \(i\).
• Consider the subgraph \(G_{i,j}\) induced by the vertices colored \(i\) and \(j\).
• If \(v_{i}\) and \(v_{j}\) are in different components, we can switch colors in one of them.

• Now, suppose that for each pair \((i,j)\), the vertices \(v_{i}\) and \(v_{j}\) are in the same component of \(G_{i,j}\).
• Let \(P_{i,j}\) be a path from \(v_{i}\) to \(v_{j}\) in \(G_{i,j}\). 
• Since we listed the vertices in order, the path \(P_{1,3}\) cuts off \(v_{2}\) from \(v_{4}\). 
• More precisely, form the cycle by adding \(v_{0}\) to \(P_{1,3}\). This forms a face.
• By the ordering of the vertices \(v_{2}\) and \(v_{4}\) cannot both be inside or outside the face.
• Thus, the path \(P_{2,4}\) must cross \(P_{1,3}\).
• This contradicts the assumption that \(G\) is planar

\(P_{1,3}\)

• Thus there's a pair \((i,j)\) such that \(v_{i}\) and \(v_{j}\) must be in different components of \(G_{i,j}\).
• Switching colors on one of those components frees up a color to color \(v_{0}\).
• Thus, we have the desired \(5\)-coloring. \(\Box\)

The four color theorem

Theorem 6.3.6. Every planar graph is \(4\)-colorable.

An application graph coloring