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.