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.

On the excessive[m]-index of a tree

“…The exchange of edges in P increases |F i | by one and decreases |F j | by one. The iteration of this process furnishes a [4]-cover of G of size s. We prove in Theorem 3 that each tree having a [4]-compatible subtree is itself compatible. Note that this property is false in general for an arbitrary graph, for instance the Petersen graph minus an edge is [5]-compatible but the Petersen graph is not.…”

“…Furthermore, an [m]-cover F of G can be viewed as a multicoloring C of the edge-set of G. More precisely, we mean that each [m]-matching of F is a color of C and so each edge e ∈ E(G) receives a number of colors equal to the number of [m]-matchings which contain e. We will say that C is the multicoloring associated to the [m]-cover F . It was useful in [4] and [5] to consider edge colorings whose color classes have approximately the same size. The same idea turned up to be useful in our proofs as well.…”

“…When the cardinality m is clear from the context, we will use the notation S in place of S(m). We will denote by s(G) the cardinality of the largest splitting set of G. The concept of a splitting set was introduced in [4] in order to produce a formula for the computation of the excessive [3]-index of an arbitrary graph. More precisely, Cariolaro and Fu proved the following theorem.…”

“…In the same way one can check that if a caterpillar T does not contain the two caterpillars in Figure 4 and it has either three or four d i 's greater than 2, then T In the following proofs we will use the fact that all trees in Figure 5 are [4]-compatible (for each tree we exhibit a [4]-cover which uses a number of [4]matchings which is equal to the maximum degree of the tree). ii) Furthermore, we would like to recall that the chromatic index χ ′ (T ) of a tree is equal to its maximum degree ∆(T ).…”

“…Furthermore, we exhibit a graph (not a tree) for which such kind of formula does not work. Finally, we conjecture a general formula for the excessive [m]-index of any tree (for all values of m) and for the excessive [4]-index of a general graph.…”