Year: 2026
Type: preprint
Abstract: A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $Δ_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\le d\leΔ$, every finite simple $d$-degenerate graph $G$ with $Δ(G)\leΔ$ satisfies $$q_B(G)\le Δ+(d-1)Δ_2(G)\le dΔ.$$ Consequently, $dΔ$ is the exact maximum, with equality precisely for graphs containing $K_{d,Δ}$. More generally, if $q_B(G)\ge dΔ-s$, where $0\le s<Δ$, then $G$ contains $K_{d,Δ-s}$; if also $s
Source: arXiv (Cornell University)
Authors: Hu, Xiaoxue, Kong, Jiangxu, Wang, Yiqiao
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G7595905878, G8631368005
OpenAlex ID: W7207857635