The equivalence number of a line graph. (English) Zbl 1268.05171
Summary: 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.
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.
MSC:
| 05C76 | Graph operations (line graphs, products, etc.) |
| 05A05 | Permutations, words, matrices |
| 05C15 | Coloring of graphs and hypergraphs |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
References:
| [1] | M. O. Albertson, R. E. Jamison, S. T. Hedetniemi, and S. C. Locke, The subchromatic number of a graph , Discrete Math., 74 (1989), 33-49. · Zbl 0681.05033 · doi:10.1016/0012-365X(89)90196-9 |
| [2] | N. Alon, Covering graphs by the minimum number of equivalence relations , Combinatorica, 6.3 (1986), 201-206. · Zbl 0646.05053 · doi:10.1007/BF02579381 |
| [3] | G. Chartrand and P. Zhang, Chromatic Graph Theory , Chapman & Hall, CRC Press, Boca Raton, FL, 2009. · Zbl 1169.05001 |
| [4] | R. D. Chatham, G. H. Fricke, J. Harless, R. D. Skaggs, and N. Wahle, Transit graphs , Preliminary Report (1033-05-15), AMS Fall Southeastern Sectional Meeting, Nov. 4, 2007. |
| [5] | M. Chudnovsky and P. Seymour, Claw-free graphs. IV. Decomposition theorem , J. Combin. Theory Ser. B, 98 (2008), 839-938. · Zbl 1152.05038 · doi:10.1016/j.jctb.2007.06.007 |
| [6] | R. Diestel, Graph Theory , 2nd ed., Springer-Verlag, New York, 2000. |
| [7] | P. Duchet, Représentations, noyaux en théorie des graphes et hypergraphes , Thése d’Etat, Université Paris VI, 1979. |
| [8] | L. Esperet, J. Gimbel, and A. King, Covering line graphs with equivalence relations , Discrete Applied Mathematics, 158 (2010), 1902-1907. · Zbl 1215.05130 · doi:10.1016/j.dam.2010.08.012 |
| [9] | Z. Furedi, On the Prague dimension of Kneser graphs , Numbers, Information and Complexity (Bielefeld, 1998) pp. 125-128, Kluwer Academic Publishers, Boston, MA, 2000. · Zbl 0946.05083 |
| [10] | T. Jensen and B. Toft, Graph Coloring Problems , John Wiley & Sons, New York, 1995. · Zbl 0855.05054 |
| [11] | L. Lovász, J. Nešetřil, and A. Pultr, On a product dimension of graphs , J. Combin. Theory Ser. B, 29.1 (1980), 47-67. · Zbl 0439.05038 · doi:10.1016/0095-8956(80)90043-X |
| [12] | C. McClain, Edge colorings of graphs and multigraphs , doctoral dissertation, The Ohio State University, 2008. |
| [13] | C. McClain, The clique chromatic index of line graphs , unpublished manuscript, 2009. |
| [14] | J. Nešetřil and A. Pultr, A Dushnik-Miller type dimension of graphs and its complexity , in Fundamentals of Computation Theory, Proc. Conf. Poznań-Kórnik, 1977, Springer Lecture Notes in Comp. Sci., 56 (1977), 482-493. · Zbl 0362.05073 |
| [15] | D. B. West, Introduction to Graph Theory , 2nd ed., Prentice Hall, Upper Saddle River, NJ, 2000. · Zbl 0989.81101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
