Planar Graphs with Ore-degree at Most seven is strongly $13$-edge-colorable

Abstract

A strong edge-coloring of a graph $G$ is a coloring of edges of $G$ such that every color class forms an induced matching. The strong chromatic index is the minimum number of colors needed to color the graph. The Ore-degree $\theta(G)$ of a graph $G$ is the maximum sum of degrees of adjacent vertices. We show that every planar graph $G$ with $\theta(G)\le 7$ has strong chromatic index at most $13$. This settles a conjecture of Chen et al in the planar case. We use a discharging method, and apply Combinatorial Nullstellensatz to show reducible configurations. We provide an algorithm to allow Combinatorial Nullstellansatz extracting coefficients from large polynomials.