Approximations for the Two-Machine Cross-Docking Flow Shop Problem
Discrete Applied Mathematics
| Hi-index | 0.00 |
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 \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 00 such that the following holds. Let G be a balanced bipartite graph with maximum degree Δ≥Δ0 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