DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 15(1) (1995) 89-106
DOI: 10.7151/dmgt.1010

SOME MAXIMUM MULTIGRAPHS AND EDGE/VERTEX DISTANCE COLOURINGS

Zdzisław Skupień

Institute of Math. AGH
al. Mickiewicza 30, 30-059 Krakಷ, Poland
e-mail: skupien@uci.agh.edu.pl

Abstract

Shannon-Vizing-type problems concerning the upper bound for a distance chromatic index of multigraphs G in terms of the maximum degree Δ(G) are studied. Conjectures generalizing those related to the strong chromatic index are presented. The chromatic d-index and chromatic d-number of paths, cycles, trees and some hypercubes are determined. Among hypercubes, however, the exact order of their growth is found.

Keywords: (strong) chromatic index, chromatic number, matching, hypercube, error-correcting code, asymptotics

1991 Mathematics Subject Classification: 05C15, 05C35, 94B05

References

[1] L.D. Andersen, The strong chromatic index of a cubic graph is at most 10, in: J. Nesetril, ed., Topological, Algebraical and Combinatorial Structures; Frolik's Memorial Volume. Discrete Math. 108 (1992) 231-252; reprinted in Vol. 8 of Topics in Discrete Math. (Elsevier, 1993).
[2] J.C. Bermond, J. Bond, M. Paoli and C. Peyrat, Graphs and interconnection networks: diameter and vulnerability, in: Surveys in Combinatorics, Proc. Ninth British Combin. Conf. (London Math. Soc., Lect. Notes Series 82, 1983) 1-30.
[3] F.R.K. Chung, A. Gyarfas, Zs. Tuza and W.T. Trotter, The maximum number of edges in 2K2-free graphs of bounded degree, Discrete Math. 81 (1990) 129-135, doi: 10.1016/0012-365X(90)90144-7.
[4] R.J. Faudree, A. Gyarfas, R.H. Schelp and Zs. Tuza, Induced matchings in bipartite graphs, Discrete Math. 78 (1989) 83-87, doi: 10.1016/0012-365X(89)90163-5.
[5] R.J. Faudree, R.H. Schelp, A. Gyarfas and Zs. Tuza, The strong chromatic index of graphs, Ars Combinat. 29-B (1990) 205-211.
[6] P. Horák, H. Qing and W.T. Trotter, Induced matchings in cubic graphs, J. Graph Theory 17 (1993) 151-160. [Communicated at 1991 Conf. in Zemplinska Sirava (CS).], doi: 10.1002/jgt.3190170204
[7] F. Kramer, Sur le nombre chromatique K(p,G) des graphes, Rev. Fran८ Automat. Inform. Rech. Opਲat. 6 (1972) 67-70; Zbl. 236,05105
[8] F. Kramer and H. Kramer, On the generalized chromatic number, in: Combinatorics '84, Proc. Int. Conf. Finite Geom. Comb. Struct., Bari/Italy, 1984 (Ann. Discrete Math. 30, 1986) 275-284; Zbl. 601,05020.
[9] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam et al., 1981).
[10] C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148-151.
[11] E. Sidorowicz and Z. Skupień A joint article in preparation.
[12] Z. Skupień Some maximum multigraphs and chromatic d-index, in: U. Faigle and C. Hoede, eds., 3rd Twente Workshop on Graphs and Combinatorial Optimization (Fac. Appl. Math. Univ. Twente, Enschede, 1993) 173-175.
[13] V.G. Vizing, Chromatic class of a multigraph [Russian], Kibernetika 3 (1965) 29-39.
[14] S. Wagon, (Note) A bound on the chromatic number of graphs without certain induced subgraphs, J. Combin. Theory (B) 29 (1978) 345-346, doi: 10.1016/0095-8956(80)90093-3.