New Proof of Adjacency Lemmas of Edge Critical Graphs

Abstract

A graph $G$ with maximum degree $\mathrm{\Delta }$ and edge chromatic number ${\chi }^{\prime }(G)>\mathrm{\Delta }$ is \emph{edge-$\mathrm{\Delta }$-critical} if ${\chi }^{\prime }(G-e)=\mathrm{\Delta }$ for each $e\in E(G)$. In this article, we provide a new proof of adjacency Lemmas on edge-critical graphs such that Vizing’s adjacency lemma becomes a corollary of our results.