Table of Contents
International Journal of Combinatorics
Volume 2015 (2015), Article ID 201427, 7 pages
http://dx.doi.org/10.1155/2015/201427
Research Article

On Evenly-Equitable, Balanced Edge-Colorings and Related Notions

Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, Al 36849-5310, USA

Received 23 September 2014; Accepted 16 February 2015

Academic Editor: Toufik Mansour

Copyright © 2015 Aras Erzurumluoğlu and C. A. Rodger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A graph is said to be even if all vertices of have even degree. Given a -edge-coloring of a graph , for each color let denote the spanning subgraph of in which the edge-set contains precisely the edges colored . A -edge-coloring of is said to be an -edge-coloring if for each color , is an even graph. A -edge-coloring of is said to be evenly-equitable if for each color , is an even graph, and for each vertex and for any pair of colors , . For any pair of vertices let be the number of edges between and in (we allow , where denotes a loop incident with ). A -edge-coloring of is said to be balanced if for all pairs of colors and and all pairs of vertices and (possibly ), . Hilton proved that each even graph has an evenly-equitable -edge-coloring for each . In this paper we extend this result by finding a characterization for graphs that have an evenly-equitable, balanced -edge-coloring for each . Correspondingly we find a characterization for even graphs to have an evenly-equitable, balanced 2-edge-coloring. Then we give an instance of how evenly-equitable, balanced edge-colorings can be used to determine if a certain fairness property of factorizations of some regular graphs is satisfied. Finally we indicate how different fairness notions on edge-colorings interact with each other.