Balanced coloring of bipartite graphs

Given a bipartite graph G(U ∪ V, E) with n vertices on each side, an independent set I ∈ G such that |U ⋂ I| = |V ⋂ I| is called a balanced bipartite independent set. A balanced coloring of G is a coloring of the vertices of G such that each color class induces a balanced bipartite independent set in G. If graph G has a balanced coloring we call it colorable. The coloring number χB(G) is the minimum number of colors in a balanced coloring of a colorable graph G. We shall give bounds on χB(G) in terms of the average degree d of G and in terms of the maximum degree ∆ of G. In particular we prove the following: • χB(G) ≤ max{2, b2dc+ 1}. • For any 0 < 2 < 1 there is a constant ∆0 such that the following holds. Let G be a balanced bipartite graph with maximum degree ∆ ≥ ∆0 and n ≥ (1+ 2)2∆ vertices on each side, then χB(G) ≤ 20 22 ∆ ln ∆ .