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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2014 | 34 | 2 | 391-408

## On eulerian irregularity in graphs

EN

### Abstrakty

EN
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

EN

391-408

wydano
2014-05-01
online
2014-04-12

### Twórcy

autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA

### Bibliografia

• [1] E. Andrews, G. Chartrand, C. Lumduanhom and P. Zhang, On Eulerian walks in graphs, Bull. Inst. Combin. Appl. 68 (2013) 12-26.
• [2] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs: 5th Edition (Chapman & Hall/CRC, Boca Raton, FL, 2010).
• [3] L. Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Academiae Sci. I. Petropolitanae 8 (1736) 128-140.
• [4] M.K. Kwan, Graphic programming using odd or even points, Acta Math. Sinica 10 (1960) 264-266 (in Chinese), translated as Chinese Math. 1 (1960) 273-277.