An improvement to the vertex-splitting conjecture

TL;DR

This paper proves a conjecture about the criticality of certain overfull graphs obtained by splitting vertices in regular graphs, extending previous results to graphs with maximum degree at least 75% of the number of vertices.

Contribution

The paper confirms Hilton and Zhao's conjecture for connected regular graphs with maximum degree at least 0.75 times the number of vertices, improving previous bounds.

Findings

Confirmed the conjecture for $oldsymbol{ riangle(G) ext{ ≥ } 0.75n}$.

Extended the range of graphs for which the conjecture holds.

Provided new insights into the structure of overfull and edge-chromatic critical graphs.

Abstract

For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$ of $G$. Define $G$ to be overfull if $|E(G)|>\Delta(G) \lfloor n/2 \rfloor$. Clearly, overfull graphs are class 2 and any graph obtained from a regular graph of even order by splitting a vertex is overfull. Let $G$ be an $n$-vertex connected regular class 1 graph with $\Delta(G) >n/3$. Hilton and Zhao in 1997 conjectured that if $G^*$ is obtained from $G$ by splitting one vertex of $G$ into two vertices, then $G^*$ is edge-chromatic critical, and they verified the conjecture for graphs $G$ with $\Delta(G)\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. The graph $G^*$ is easily verified to be overfull, and so the hardness of the conjecture lies in showing that the deletion of every of its edge decreases the chromatic index. Except in 2002, Song showed that the conjecture is true for a special class of graphs $G$ with $\Delta(G)\ge \frac{n}{2}$, no other progress on this conjecture had been made. In this paper, we confirm the conjecture for graphs $G$ with $\Delta(G) \ge 0.75n$.