On strongly spanning $k$-edge-colorable subgraphs

TL;DR

This paper introduces a new graph parameter $sp(G)$ related to strongly spanning maximum $k$-edge-colorable subgraphs, providing bounds, characterizations, and alternative definitions, advancing understanding of edge-coloring properties.

Contribution

The paper defines the parameter $sp(G)$, characterizes graphs where $sp(G)$ equals maximum degree, and establishes bounds involving classical graph parameters, offering new insights into edge-coloring.

Findings

$sp(G)$ is bounded above by $ riangle(G)$

Graphs with $sp(G)= riangle(G)$ are characterized

Bounds for $sp(G)$ involve well-known graph parameters

Abstract

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a graph-parameter $sp(G)$, that coincides with the smallest $k$ that a graph $G$ has a strongly spanning maximum $k$-edge-colorable subgraph. Our first result offers some alternative definitions of $sp(G)$. Next, we show that $\Delta(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=\Delta(G)$. Finally, we prove some bounds for $sp(G)$ that involve well-known graph-theoretic parameters.