List Strong Edge-Colorings of Sparse Graphs

A strong edge-coloring of a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} is a partition of its edge set E into induced matchings. In this paper, we will study the list version of strong edge-colorings of several classes of sparse graphs, including bipartite graphs and graphs with small edge weight, where the edge weight of a graph is defined by max{dG(u)+dG(v)|uv∈E(G)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \{ d_G(u) + d_G(v)| uv \in E(G)\}$$\end{document}. We show that: (1) if G is a bipartite graph with bipartition (A, B) such that Δ(A)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (A)=2$$\end{document} and Δ(B)=Δ≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (B)=\Delta \ge 4$$\end{document}, then G has strong list-chromatic index at most 3Δ-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\Delta -3$$\end{document}; (2) every graph with edge weight at most 5 (resp. 6) has strong list-chromatic index at most 7 (resp. 11) and every planar graph with edge weight at most 6 has strong list-chromatic index at most 10; and (3) every graph with edge weight at most 7 has strong list-chromatic index at most 16.