On the excessive[m]-index of a tree

The excessive m ] -index of a graph G , denoted by ? m ] ' ( G ) , is the minimum number of matchings of size m needed to cover the edge-set of G . We set ? m ] ' ( G ) = ∞ if such a cover does not exist and we call a graph G m ] -coverable if its excessive m ] -index is finite. Obviously ? 1 ] ' ( G ) = | E ( G ) | and it is easy to prove for a 2 ] -coverable graph G that ? 2 ] ' ( G ) = max { ? ' ( G ) , ? | E ( G ) | / 2 ? } holds, where ? ' ( G ) denotes the chromatic index of G . The case m = 3 is completely solved in Cariolaro and Fu (2009)? 4]. In this paper we prove a general formula for computing the excessive 4 ] -index of a tree and we conjecture a possible generalization for any value of m . Furthermore, we prove that such a formula does not work for the excessive 4 ] -index of an arbitrary graph.