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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.

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A petal graph is a connected graph $G$ with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a $2$–regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class $1$. This settles a particular case of a conjecture of Hilton and Zhao.

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