Strong edge-coloring of sparse graphs with Ore-degree 7 or 8
Runze Wang
TL;DR
The paper addresses strong edge-coloring in graphs with respect to Ore-degree, aiming to bound the strong chromatic index under sparsity constraints. It combines discharging methods with Hall's marriage theorem to prove two main results: (i) if $\theta(G)\le 7$ and $\mathrm{mad}(G)<\frac{34}{11}$, then $\chi_s'(G)\le 13$, improving the prior bound, and (ii) if $\theta(G)\le 8$ and $\mathrm{mad}(G)<\frac{113}{31}$, then $\chi_s'(G)\le 20$. Both proofs proceed via assuming a minimal counterexample, deriving a detailed vertex-structure classification, and constructing a charge-discharge scheme to reach a contradiction, with Hall-based color-extension steps ensuring feasible color assignments. The results push toward Chen et al.'s conjecture for Ore-degree-based bounds in sparse graphs and demonstrate the effectiveness of combining discharging with matching techniques in strong edge-coloring problems.
Abstract
In a strong edge-coloring of a graph $G=(V,E)$, any two edges of distance at most $2$ get distinct colors. The strong chromatic index of $G$, denoted by $χ_s'(G)$, is the minimum number of colors needed in a strong edge-coloring of $G$. The Ore-degree of $G$ is defined by $\max\{d(u)+d(v):uv\in E\}$. In this paper, we apply the discharging method and make use of Hall's marriage theorem to prove two results toward a conjecture by Chen et al. First, we prove that if $G$ is a graph with Ore-degree $7$ and maximum average degree less than $\frac{34}{11}$, then $χ_s'(G)\le 13$. This result improves the previous best bound from $\frac{40}{13}$ to $\frac{34}{11}$. Second, we prove that if $G$ is a graph with Ore-degree $8$ and maximum average degree less than $\frac{113}{31}$, then $χ_s'(G)\le 20$.