A Note on the Facial Edge-Coloring Conjecture

Let G be a connected plane graph that can have loops and multiple edges. An l-facial edge-coloring of a plane graph G is a coloring of edges of G such that any two edges, that share the same facial trail of length at most l+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l + 1$$\end{document}, receive distinct colors. It is an edge variant of the l-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It was conjectured by Lu & zcaron;ar et al. in 2015 that every plane graph admits an l-facial edge-coloring with at most 3l+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3l + 1$$\end{document} colors for any l >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l \ge 1$$\end{document}. It is known that the bound 3l+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3l+1$$\end{document} is tight for general plane graphs. The conjecture was recently confirmed for l <= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l \le 3$$\end{document} by Hor & ncaron;& aacute;k, Lu & zcaron;ar and & Scaron;torgel (3-facial edge-coloring of plane graphs, Discrete Math. 346 (2023) 113312). In this note we prove that the conjecture holds, in the case when l >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l \ge 4$$\end{document}, for every graph whose reduction (the graph obtained from G by suppressing all its 2-vertices) is 3-edge connected, and the length of the longest path in G with interior vertices of degree 2 is at most 3l+110\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3l + 1}{10}$$\end{document}.