Excessive [m]-index Of Some Special Graphs

A set F of matchings of G, all of which of size m, is called an [m]-cover of G if∪M ∈FM = E(G). An [m]-cover F such that |F | is minimum and the number of matchings it contains is a graph parameter called excessive [m]-index and denoted byχ′[m](G). If [m]-matching is perfect matching or near-perfect matching, the corresponding parameter called excessive index of G, denoted by χ′e(G). For small values of m, triviallyχ′[1](G) = |E(G)| holds and χ′[2](G) = max{χ′(G), |E(G)|/2} holds for each [2]-coverable graph, where χ′(G) denotes the chromatic index of G. Cariolaro and Fu have proved the following formula χ′[3](G) = max{χ′(G), |E(G)|/3, s(G)} holds for every [3]-coverable graph, where s(G) denotes the splitting index of G. Furthermore, they have proved the following formula χ′[3](T) = max{χ′(T), |E(T)|/3} holds for all [3]-coverable trees.For every [4]-coverable tree, the formula has been proved by G.Mazzuoccolo χ′[4](T) =max{χ′(T), |E(T)|/4, s(T)}. However the task of determining χ′[m](G) for arbitrary m and G seems to increase very rapidly in diculty as m increases. The particular case m =|V(G)|2is largely studied. Rajasingh et al. have determined excessive index for hexagonal and 3-D mesh network, for mesh, cylinder and torus networks.We further compute the excessive [m]-index for some special families of graphs, such as unicyclic graphs, Cartesian product and Direct product of some graphs.