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A Stronger Bound for the Strong Chromatic Index

  • HENNING BRUHN (a1) and FELIX JOOS (a1)

Abstract

We prove χ′ s (G) ≤ 1.93 Δ(G)2 for graphs of sufficiently large maximum degree where χ′ s (G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.

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[1] Alon, N., Krivelevich, M. and Sudakov, B. (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 7382.
[2] Andersen, L. D. (1992) The strong chromatic index of a cubic graph is at most 10. Discrete Math. 108 231252.
[3] Azuma, K. (1967) Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357367.
[4] Bonamy, M., Perrett, T. and Postle, L. (2016) Reed's conjecture and strong edge coloring. Paper given at Scottish Combinatorics Meeting, Glasgow.
[5] Chung, F. R. K., Gyárfás, A., Tuza, Z. and Trotter, W. T. (1990) The maximum number of edges in 2K2-free graphs of bounded degree. Discrete Math. 81 129135.
[6] Cranston, D. W. (2006) Strong edge-coloring of graphs with maximum degree 4 using 22 colors. Discrete Math. 306 27722778.
[7] Diestel, R. (2010) Graph Theory, fourth edition, Springer.
[8] Faudree, R. J., Schelp, R. H., Gyárfás, A. and Tuza, Z. (1989) Induced matchings in bipartite graphs. Discrete Math. 78 8387.
[9] Faudree, R. J., Schelp, R. H., Gyárfás, A. and Tuza, Z. (1990) The strong chromatic index of graphs. Ars Combin. 29 205211.
[10] Grable, D. A. (1998) A large deviation inequality for functions of independent, multi-way choices. Combin. Probab. Comput. 7 5763.
[11] Horák, P., Qing, H. and Trotter, W. T. (1993) Induced matchings in cubic graphs. J. Graph Theory 17 151160.
[12] Joos, F. (2016) Induced matchings in graphs of bounded maximum degree. SIAM J. Discrete Math. 30 18761882.
[13] Joos, F. and Nguyen, V. H. (2016) Induced matchings in graphs of degree at most 4. SIAM J. Discrete Math. 30 154165.
[14] Joos, F., Rautenbach, D. and Sasse, T. (2014) Induced matchings in subcubic graphs. SIAM J. Discrete Math. 28 468473.
[15] Kaiser, T. and Kang, R. (2014) The distance-t chromatic index of graphs. Combin. Probab. Comput. 23 90101.
[16] Kim, J. H. (1995) On Brooks' theorem for sparse graphs. Combin. Probab. Comput. 4 97132.
[17] Kim, J. H. and Vu, V. H. (2000) Concentration of multivariate polynomials and its applications. Combinatorica 20 417434.
[18] Kutin, S. (2002) Extensions to McDiarmid's inequality when differences are bounded with high probability. Technical Report TR-2002-04, University of Chicago.
[19] Mahdian, M. (2000) The strong chromatic index of C 4-free graphs. Random Struct. Alg. 17 357375.
[20] McDiarmid, C. (1989) On the method of bounded differences. In Surveys in Combinatorics, Vol. 141 of London Mathematical Society Lecture Note series, Cambridge University Press, pp. 148188.
[21] McDiarmid, C. (2002) Concentration for independent permutations. Combin. Probab. Comput. 11 163178.
[22] Molloy, M. and Reed, B. (1997) A bound on the strong chromatic index of a graph. J. Combin. Theory Ser. B 69 103109.
[23] Molloy, M. and Reed, B. (1998) A bound on the total chromatic number. Combinatorica 18 241280.
[24] Molloy, M. and Reed, B. (2002) Graph Coloring and the Probabilistic Method, Springer.
[25] Rivin, I. (2002) Counting cycles and finite dimensional L p norms. Adv. Appl. Math. 29 647662.
[26] Śleszyńska-Nowak, M. (2016) Clique number of the square of a line graph. Discrete Math. 339 15511556.
[27] Talagrand, M. (1995) Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73205.
[28] van Lint, J. H. (1999) Introduction to Coding Theory, Graduate Texts in Mathematics, New York University Press.
[29] Warnke, L. (2016) On the method of typical bounded differences. Combin. Probab. Comput. 25 269299.

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A Stronger Bound for the Strong Chromatic Index

  • HENNING BRUHN (a1) and FELIX JOOS (a1)

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