Coloring of the graphs

Anna Powolny

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Subtitle

Graph Coloring?

Special case of graph labeling
Assignment of labels - colors - to elements of a graph

Vertex Coloring

no two vertices sharing the same edge have the same color.
Graphs are loopless. (connection never directly back to itself)

k-coloring

- coloring using at most k colors

k-colorable

- graph having k-coloring (chromatic number χ(G)<=k)

chromatic number

- χ(G) - the smallest number of colors needed to color a graph G.

k-chromatic graph

- chromatic number χ(G) = k

one-colorable

Empty graph - isolated nodes with no edges.

Bipartite Graph

Two-colorable graph.

vertices can be partioned into two sets X and Y
each edge has one vertex in X and one in Y

color class

a subset of verticies assigned to the same color, every such a class forms an independent set.

Edge Coloring

no two adjacent edges share the same color

k-edge-coloring

coloring using at most k colors

chromatic index

(edge chromatic number)

χ'(G) - the smallest number of colors needed for an edge coloring of a graph G.

Face Coloring

no two faces that share a boundary have the same color
Note the red color that goes to infinity.

Theorems

König's theorem for Bipartite Graph

χ'(G) = Δ(G)

χ'(G) - the smallest number of colors needed for an edge coloring of a graph G.

Δ(G) - maximum vertex degree

(maximum number of edges that vertex in graph G has.)

χ(G) - the smallest number of colors needed to color a graph. G.

Δ(G) - maximum vertex degree

(maximum number of edges that vertex in graph G has.)

Brooks Theorem

for a connected, simple graph G, unless G is a complete graph or an odd cycle.

Simple Graph

is an unweighted, undirected graph containing no graph loops or multiple edges

Complete Graph

is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge

Odd Cycle Graph

a cycle graph with an odd number of vertices.

The Greedy Algorithm

Choose an order for the vertices
Choose a list of colors, also in some order
In order, color each vertex using the first available color on the list, making sure that no two adjacent vertices have the same coloring
Continue in this way until each vertex is colored

Choose an order for the vertices
Choose a list of colors, also in some order

3. In order, color each vertex using the first available color on the list, making sure that no two adjacent vertices have the same coloring

4. Continue in this way until each vertex is colored

Applications

Scheduling

a given set of jobs need to be assigned to time slots
pairs of jobs may be in conflict in the sense that they may not be assigned to the same time slot
vertex for every job and an edge for every conflicting pair of jobs

The chromatic number of the graph is the optimal time to finish all jobs without conflicts

Sudoku

81 blocks as 81 verticies
digits from 1 to 9 as a 9 different colors

completing a 9-coloring on given specific graph with 81 vertices.

The End