Coloring Graphs Without Bichromatic Cycles or Paths

Let k≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 4$$\end{document} be an integer, and let G be a graph with maximum degree Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}. In 1991, Alon, McDiarmid and Reed proved that G has a proper coloring using O(Δ(k-1)/(k-2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\Delta ^{(k-1)/(k-2)})$$\end{document} colors such that G does not have bichromatic paths with k vertices. In this paper, we improve this result by showing G has a proper coloring using (1+⌈k/2⌉1/(k-3))Δ(k-1)/(k-2)+Δ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\lceil k/2\rceil ^{1/(k-3)})\Delta ^{(k-1)/(k-2)}+\Delta +1$$\end{document} colors such that G does not have bichromatic paths with k vertices. We remark that there exists a graph G with maximum degree Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} such that for any proper coloring of G using Ω(Δ(k-1)/(k-2)(logΔ)1/(k-2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\frac{\Delta ^{(k-1)/(k-2)}}{(\log \Delta )^{1/(k-2)}})$$\end{document} colors, there is always a bichromatic path with k vertices. Using the similar method, we also show that G has a proper coloring using O(Δ4/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\Delta ^{4/3})$$\end{document} colors such that G contains neither bichromatic cycles nor bichromatic paths with order 5.