Geodesic
On Eulerian irregularity in graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 391-408

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A closed walk in a connected graph G that contains every edge of G exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph G is Eulerian if and only if every vertex of G is even. An Eulerian walk in a connected graph G is a closed walk that contains every edge of G at least once, while an irregular Eulerian walk in G is an Eulerian walk that encounters no two edges of G the same number of times. The minimum length of an irregular Eulerian walk in G is called the Eulerian irregularity of G and is denoted by EI(G). It is known that if G is a nontrivial connected graph of size m, then m+12≤ EI(G) ≤ 2 m+12. A necessary and sufficient condition has been established for all pairs k, m of positive integers for which there is a nontrivial connected graph G of size m with EI(G)=k. A subgraph F in a graph G is an even subgraph of G if every vertex of F is even. We present a formula for the Eulerian irregularity of a graph in terms of the size of certain even subgraph of the graph. Furthermore, Eulerian irregularities are determined for all graphs of cycle rank 2 and all complete bipartite graphs.
Keywords: Eulerian walks, Eulerian irregularity 
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     title = {On {Eulerian} irregularity in graphs},
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Andrews, Eric; Lumduanhom, Chira; Zhang, Ping. On Eulerian irregularity in graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 391-408. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a12/