Strong edge chromatic index of subquartic graphs

Abstract

A strong k-edge coloring of a graph G is an assignment of k colors to the edges of G such that for any two edges e and f with distance at most two receive different colors. The minimum number of k such that G has a strong k-edge coloring is called the strong chromatic index of G, denoted as χs′(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi '_s(G)$$\end{document}. In this paper, we proved that for a subquartic graph G, χs′(G)≤12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _s'(G)\le 12$$\end{document} if mad(G)<145\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mad(G)<\frac{14}{5}$$\end{document}, where mad(G)=max{2|E(H)||V(H)|,H⊆G}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mad(G)=\max \{\frac{2|E(H)|}{|V(H)|},H\subseteq G\}$$\end{document}.