Missouri Journal of Mathematical Sciences

The Equivalence Number of a Line Graph

Christopher McClain

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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.

Article information

Missouri J. Math. Sci., Volume 25, Issue 1 (2013), 61-75.

First available in Project Euclid: 28 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A05: Permutations, words, matrices
Secondary: 05C15: Coloring of graphs and hypergraphs 05C70: Factorization, matching, partitioning, covering and packing

equivalence number chromatic index clique


McClain, Christopher. The Equivalence Number of a Line Graph. Missouri J. Math. Sci. 25 (2013), no. 1, 61--75. doi:10.35834/mjms/1369746398. https://projecteuclid.org/euclid.mjms/1369746398

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