On equitable coloring of bipartite graphs

Abstract

If the vertices of a graph G are partitioned into k classes V 1, V 2, …, V k such that each V i is an independent set and ‖ V i | − | V j ‖ ⩽ 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 χ e( G) of G. The Equitable Coloring Conjecture asserts that χ e( G) ⩽ Δ( 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 | X| = m ⩾ n = | Y| and the number of edges is less than ⌊ m/( n + 1)⌋( m − n) + 2 m, then we can establish an improved bound χ e ( G) ⩽ ⌈ m/( n + 1)⌉ + 1.