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May 2013 The Equivalence Number of a Line Graph
Christopher McClain
Missouri J. Math. Sci. 25(1): 61-75 (May 2013). DOI: 10.35834/mjms/1369746398

Abstract

The chromatic index of a graph $G$ is most often defined to be the minimum size of a partition of the edge set of $G$ into matchings. An equivalent but different definition is the minimum size of a cover of the edge set of $G$ by matchings. We consider the analogous problem of covering the edge set of $G$ by subgraphs that are vertex-disjoint unions of cliques, known as equivalence graphs. The minimum size of such a cover is the equivalence number of $G$. We compute the equivalence number of the line graph of a clique on at most 12 vertices. We also construct a particular type of cover to show that, for all graphs $G$ on at most n vertices, the equivalence number of the line graph of $G$ has an upper bound on the order of log n. Finally, we show that if $G$ is a clique on 13 vertices then the minimum size of this particular cover is 5.

Citation

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Christopher McClain. "The Equivalence Number of a Line Graph." Missouri J. Math. Sci. 25 (1) 61 - 75, May 2013. https://doi.org/10.35834/mjms/1369746398

Information

Published: May 2013
First available in Project Euclid: 28 May 2013

zbMATH: 1268.05171
MathSciNet: MR3087689
Digital Object Identifier: 10.35834/mjms/1369746398

Subjects:
Primary: 05A05
Secondary: 05C15, 05C70

Rights: Copyright © 2013 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.25 • No. 1 • May 2013
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