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In an undirected connected graph G = (N, A), an Euler tour is a walk that starts at some node, visits each arc exactly once, and returns to the starting node. A graph is Eulerian if it contains an Euler tour. Show that in an Eulerian graph, the degree of every node is even. Next, show that if every node in a connected graph has an even degree, the graph is Eulerian. Establish the second result by describing an O(m) algorithm for determining whether a graph is Eulerian and, if so, will construct an Euler tour. (Hint: Describe an algorithm that decomposes any graph with only even-degree nodes into a collection of arc-disjoint cycles, and then converts the cycles into an Euler tour.)
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Senior OPerations Research Scientist

Interviewed at Optym

3.9★
Dec 24, 2022

In an undirected connected graph G = (N, A), an Euler tour is a walk that starts at some node, visits each arc exactly once, and returns to the starting node. A graph is Eulerian if it contains an Euler tour. Show that in an Eulerian graph, the degree of every node is even. Next, show that if every node in a connected graph has an even degree, the graph is Eulerian. Establish the second result by describing an O(m) algorithm for determining whether a graph is Eulerian and, if so, will construct an Euler tour. (Hint: Describe an algorithm that decomposes any graph with only even-degree nodes into a collection of arc-disjoint cycles, and then converts the cycles into an Euler tour.)

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