An efficient case for computing minimum linear arboricity with small maximum degree

Graph coloring has interesting applications in optimization, calculation of Hessian matrix, network design and so on. In this paper, we consider an improper edge coloring which is one important coloring-linear arboricity. For a graph G, a linear forest is a disjoint union of paths and cycles. The linear arboricity la(G) is the minimum number of disjoint linear forests such that their union is exactly the edge set of G. In this paper, we study a special case that G is a simple planar graph with two not adjacent cycles each with a chordal and length between 4 and 7. We show that in this special case, la(G)=⌈Δ2⌉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$la(G)=\lceil \frac{\Delta }{2}\rceil $$\end{document} where 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} is the maximum vertex degree of G and Δ≥7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \ge 7$$\end{document}.