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 χ B ( G ) is the minimum number of colors in a balanced coloring of a colorable graph G . We shall give bounds on χ B ( G ) in terms of the average degree \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\bar{d}$\end{document} of G and in terms of the maximum degree Δ of G . In particular we prove the following:\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\chi_{{{B}}}({{G}}) \leq {{max}} \{{{2}},\lfloor {{2}}\overline{{{d}}}\rfloor+{{1}}\}$\end{document} . For any 0<ε<1 there is a constant Δ 0 such that the following holds. Let G be a balanced bipartite graph with maximum degree Δ≥Δ 0 and n ≥(1+ε)2Δ vertices on each side, then \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\chi_{{{B}}}({{G}})\leq \frac{{{{20}}}}{\epsilon^{{{2}}}} \frac{\Delta}{{{{ln}}}\,\Delta}$.\end{document}© 2009 Wiley Periodicals, Inc. J Graph Theory 64: 277–291, 2010