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 [...] . 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
@article{bwmeta1.element.doi-10_7151_dmgt_1744,
author = {Eric Andrews and Chira Lumduanhom and Ping Zhang},
title = {On eulerian irregularity in graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {391-408},
zbl = {1290.05091},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1744}
}
Eric Andrews; Chira Lumduanhom; Ping Zhang. On eulerian irregularity in graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 391-408. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1744/
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