Year: 2026
Type: preprint
Abstract: Let $φ$ be a proper edge-coloring of a graph $G$. An edge is called rich if all edges adjacent to it receive pairwise distinct colors. The coloring $φ$ is a rich-neighbor edge-coloring if every non-isolated edge is adjacent to a rich edge. Petruševski and Škrekovski introduced this notion as a relaxation of strong edge-coloring and conjectured that every connected subcubic graph other than $K_4$ admits a rich-neighbor $5$-edge-coloring. We establish the conjectured bound for every subcubic graph $G$ satisfying $Δ(G[V_3])\leq 1,$ where $V_3$ is the set of vertices of degree $3$ in $G$. Equivalently, the $3$-vertices of $G$ induce a matching together with isolated vertices. This extends the result of Chen, Tan and Zhou for subcubic graphs with no adjacent $3$-vertices.
Source: arXiv (Cornell University)
Authors: Kong, Jiangxu, Lin, Zhuming, Wang, Yan
Institutions: Hangzhou Normal University, Minnan Normal University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G5043163333
OpenAlex ID: W7222459893