✦ LIBER ✦

On equitable coloring of bipartite graphs

✍ Scribed by Ko-Wei Lih; Pou-Lin Wu

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 285 KB
Volume: 151
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(94)00092-w

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✦ Synopsis

If the vertices of a graph G are partitioned into k classes V~, I/2 ..... Vk such that each V~ is an independent set and I1V~I-IV~[I ~< 1 for all i#j, then G is said to be equitably colored with k colors. The smallest integer n for which G can be equitably colored with n colors is called the equitable chromatic number xc(G) of G. The Equitable Coloring Conjecture asserts that zc(G) ~ A(G) for all connected graphs G except the complete graphs and the odd cycles. We show that this conjecture is true for any connected bipartite graph G(X, Y). Furthermore, if IXI = m/> n = I YI and the number of edges is less than Lm/(n + 1)J(m --n) + 2m, then we can establish an improved bound Xo (G) ~< Fm/(n + 1)-] + 1.

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