**Euler Tours**

Neeldhara Misra

# DSA1 Week 4

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Work through the exercises in the Mathigon lesson

about the Bridges of Königsberg

A **walk** is a sequence of edges

\(e_1, \ldots, e_{n-1}\)

such that there exists a sequence of vertices

\(v_1, \ldots, v_n\)

for which \(e_i = (v_i, v_{i+1})\)

for all \(1 \leqslant i \leqslant n-1\).

An (closed/open) **Euler Tour **of a graph \(G\)

is a (closed/open) walk

that contains every edge *exactly *once

(i.e, no repeats and no omissions).

If a graph \(G\) has:

(a) more than two vertices of odd degree, or

(b) exactly one vertex of odd degree, then

it **does not** have an Euler Tour of any kind

(closed or open).

**What we discovered**

If a graph \(G\) has:

(a) exactly two vertices of odd degree, or

(b) no vertex of odd degree, then

**does it** have an Euler Tour of *some* kind

(closed or open)?

**Food for thought**

#### 2023 DSA1 | Week 4

By Neeldhara Misra

# 2023 DSA1 | Week 4

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