The Excessive [3]-Index of All Graphs

Abstract

Let $m$ be a positive integer and let $G$ be a graph. A set ${\cal M}$ of matchings of $G$, all of which of size $m$, is called an $[m]$-covering of $G$ if $\bigcup_{M\in {{\cal M}}}M=E(G)$. $G$ is called $[m]$-coverable if it has an $[m]$-covering. An $[m]$-covering ${\cal M}$ such that $|{{\cal M}}|$ is minimum is called an excessive $[m]$-factorization of $G$ and the number of matchings it contains is a graph parameter called excessive $[m]$-index and denoted by $\chi'_{[m]}(G)$ (the value of $\chi'_{[m]}(G)$ is conventionally set to $\infty$ if $G$ is not $[m]$-coverable). It is obvious that $\chi'_{[1]}(G)=|E(G)|$ for every graph $G$, and it is not difficult to see that $\chi'_{[2]}(G)=\max\{\chi'(G),\lceil |E(G)|/2 \rceil\}$ for every $[2]$-coverable graph $G$. However the task of determining $\chi'_{[m]}(G)$ for arbitrary $m$ and $G$ seems to increase very rapidly in difficulty as $m$ increases, and a general formula for $m\geq 3$ is unknown. In this paper we determine such a formula for $m=3,$ thereby determining the excessive $[3]$-index for all graphs.