The excessive [m]-index of a graph G, denoted by χ′[m](G), is the minimum number of matchings of size m needed to cover the edge-set of G. We set χ′[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 χ′(G) = |E(G)| and it is easy to prove for a -coverable graph G that χ′(G) = max {χ′(G), ⌈|E(G)|/2⌉} holds, where χ′(G) denotes the chromatic index of G. The case m = 3 is completely solved in Cariolaro and Fu (2009) . In this paper we prove a general formula for computing the excessive -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 -index of an arbitrary graph.

On the excessive [m]-index of a tree / Mazzuoccolo, Giuseppe. - In: DISCRETE APPLIED MATHEMATICS. - ISSN 0166-218X. - 162:(2014), pp. 264-270. [10.1016/j.dam.2013.08.014]

### On the excessive [m]-index of a tree

#### Abstract

The excessive [m]-index of a graph G, denoted by χ′[m](G), is the minimum number of matchings of size m needed to cover the edge-set of G. We set χ′[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 χ′(G) = |E(G)| and it is easy to prove for a -coverable graph G that χ′(G) = max {χ′(G), ⌈|E(G)|/2⌉} holds, where χ′(G) denotes the chromatic index of G. The case m = 3 is completely solved in Cariolaro and Fu (2009) . In this paper we prove a general formula for computing the excessive -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 -index of an arbitrary graph.
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On the excessive [m]-index of a tree / Mazzuoccolo, Giuseppe. - In: DISCRETE APPLIED MATHEMATICS. - ISSN 0166-218X. - 162:(2014), pp. 264-270. [10.1016/j.dam.2013.08.014]
Mazzuoccolo, Giuseppe
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11380/1310840`
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