On eulerian irregularity in graphs

• Autorzy: Eric Andrews , Chira Lumduanhom , Ping Zhang
• Języki publikacji: 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

Słowa kluczowe

EN

Eulerian walks; Eulerian irregularity

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 2
• Strony: 391-408

Daty

wydano

2014-05-01

online

2014-04-12

Twórcy

autor

Eric Andrews

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

autor

Chira Lumduanhom

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

autor

Ping Zhang

• 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.

Identyfikatory

DOI

10.7151/dmgt.1744