Maximal $k$-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture

TL;DR

This paper proves a new property of maximal k-edge-colorable subgraphs in multigraphs, leading to implications for Vizing's Theorem and a special case of Tuza's Conjecture on triangle packing and covering.

Contribution

It introduces a novel inequality relating degrees in maximal k-edge-colorable subgraphs, providing new proofs and extensions of classical theorems in graph theory.

Findings

Proves a degree inequality for vertices in maximal k-edge-colorable subgraphs.

Derives Vizing's Theorem as a corollary of the main result.

Establishes a special case of Tuza's Conjecture on triangle packing and covering.

Abstract

We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \{v \in V(G) : d_M(v) \leq k-\mu(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.