Euler Tours, Trail

Euler Tour → Tour that traverses each edge exactly once. Euler Trail → Trail that traverses each edge exactly once.

Closed Walk

$[u, v]$-walk in $G$ in which $u=v$

Tour, Trail

Tour → Closed walk that traverses all edges. Trail → Walk that traverses each edge at most once.

theorem #1

A connected graph $G$ has an Euler Tour $\Leftrightarrow$ each of its vertexes has an even degree. Proof → Proof is done by parts

theorem #2

If $G$ is a given graph in which the degree of each vertex is at least 2, then $G$ contains a cycle.

theorem #3

A connected graph $G$ with at least 2 vertexes has an (non-closed) Euler trail $\Leftrightarrow$ it has exactly two vertexes of odd degree. Moreover, the trail starts and ends in the vertexes of odd degree. Proof → Proof is done by parts.

Fleury’s Algorithm

Given a connected graph $G$ in which all vertexes have an even degree, we can use Fleury’s Algorithm to compute an Euler tour in $G$.

  1. Select an arbitrary vertex $u\in G$ and then set $T=\{u\}$ ($T$ will be expanded until it becomes an Euler tour in $G$)

  2. While $E(T)\neq E(G)$, do the following:

    1. Let $v$ be the last vertex added to $T$. Choose an edge $\langle v, w\rangle$ in $G-E(T)$ and add it to $T$

    Note - If $\langle v, w\rangle$ is a cut edge in $G-E(T)$ then it will be selected by the algorithm if there is no other option

    Applied Example

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