The Excessive [3]-Index of All Graphs

Let m be a positive integer and let G be a graph. A set M of matchings of G, all of which of size m, is called an [m]-covering of G if ⋃ M∈MM = E(G). G is called [m]-coverable if it has an [m]-covering. An [m]-covering M such that |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 χ[m](G) (the value of χ[m](G) is conventionally set to ∞ if G is not [m]-coverable). It is obvious that χ[1](G) = |E(G)| for every graph G, and it is not difficult to see that χ[2](G) = max{χ ′(G), ⌈|E(G)|/2⌉} for every [2]-coverable graph G. However the task of determining χ[m](G) for arbitrary m and G seems to increase very rapidly in difficulty as m increases, and a general formula for m > 3 is unknown. In this paper we determine such a formula for m = 3, thereby determining the excessive [3]-index for all graphs.