DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

DOI 10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2017: 0.601

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 22(1) (2002) 173-182
DOI: 10.7151/dmgt.1166

THREE EDGE-COLORING CONJECTURES

Richard H. Schelp

Department of Mathematical Sciences
University of Memphis
Memphis, TN 38152, USA
e-mail: rschelp@memphis.edu

Abstract

The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.

Keywords: edge-coloring, Ramsey number, vertex-distinguishing edge-coloring, strong chromatic index, balanced edge-coloring, local coloring, mean coloring.

2000 Mathematics Subject Classifications: 05C15, 05C55, 05C78.

References

[1] M. Aigner, E. Triesch and Zs. Tuza, Irregular assignments and vertex-distinguishing edge-colorings, in: Combinatorics 90, A. Barlotti et al, eds. (Elsevier Science Pub., New York, 1992), 1-9.
[2] P.N. Balister, Packing circuits in Kn, Combinatorics, Probability and Computing 10 (2001) 463-499, doi: 10.1017/S0963548301004771.
[3] P.N. Balister, B. Bollobás and R.H. Schelp, Vertex-distinguishing colorings of graphs with Δ(G) = 2, (to appear in Discrete Mathematics).
[4] P.N. Balister, A. Kostochka, Hao Li and R.H. Schelp, Balanced edge colorings, preprint.
[5] P.N. Balister, O.M. Riordan, R.H. Schelp, Vertex-distinguishing edge colorings of graphs, (to appear in J. Graph Theory).
[6] C. Bazgan, A. Harket-Benhamdine, Hao Li and Mariusz Woźniak, On the vertex-distinguishing proper edge-colorings of graphs, J. Combin. Theory (B) 74 (1999) 288-301, doi: 10.1006/jctb.1998.1884.
[7] B. Bollobás, A. Kostochka and R.H. Schelp, Local and mean Ramsey numbers for trees, J. Combin. Theory (B) 79 (2000) 100-103, doi: 10.1006/jctb.2000.1950.
[8] A.C. Burris, Vertex-distinguishing edge-colorings, Ph.D. Dissertation (Memphis State University, August 1993).
[9] A.C. Burris and R.H. Schelp, Vertex-distinguishing proper edge-colorings, J. Graph Theory 26 (2) (1997) 73-82, doi: 10.1002/(SICI)1097-0118(199710)26:2<73::AID-JGT2>3.0.CO;2-C.
[10] P. Erdős and V.T. Sós, Some Remarks on Ramsey's and Turán's theorems, in: P. Erdoos et al eds., Comb. Theory and Appl., Proc. Colloq. Math. Soc. János Bolyai 4, Balatonfüred, 1969 (North-Holland, Amsterdam, 1970) 395-404.
[11] Y. Caro, On several variations of the Turan and Ramsey numbers, J. Graph Theory 16 (1992) 257-266, doi: 10.1002/jgt.3190160309.
[12] Y. Caro and Zs. Tuza, On k-local and k-mean colorings of graphs and hypergraphs, Quart. J. Math. Oxford 44 (2) (1993) 385-398, doi: 10.1093/qmath/44.4.385.
[13] J. Cerńy, M. Hornák and R. Soták, Observability of a graph, Math. Slovaca 46 (1996) 21-31.
[14] R.A. Clapsadle, Polychromatic structures and substructures in edge-colorings of graphs, Ph. D. Dissertation (University of Memphis, 1994).
[15] R.A. Clapsadle and R.H. Schelp, Local edge-colorings that are global, J. Graph Theory 18 (1994) 389-399, doi: 10.1002/jgt.3190180409.
[16] O. Favaron, Hao Li and R.H. Schelp, Strong edge-colorings of graphs, Discrete Math. 159 (1996) 103-109, doi: 10.1016/0012-365X(95)00102-3.
[17] A. Galluccio, M. Simonovits and G. Simonyi, On the structure of co-critical graphs, in: Proc. of Graph Theory, Combinatorics, and Computing (Kalamazoo, MI) 2 (1995) 1053-1071.
[18] A. Gyárfás, J. Lehel, R.H. Schelp and Zs. Tuza, Ramsey numbers for local colorings, Graphs and Combinatorics 3 (1987) 267-277, doi: 10.1007/BF01788549.
[19] M. Hornák and R. Soták, Observability of complete multipartite graphs with equipotent parts, Ars Combin. 41 (1995) 289-301.
[20] M. Hornák and R. Soták, Asymptotic behavior of the observability of Qn, preprint.
[21] R.H. Schelp, Local and mean k-Ramsey numbers for the complete graph, J. Graph Theory 24 (1997) 201-203, doi: 10.1002/(SICI)1097-0118(199703)24:3<201::AID-JGT1>3.0.CO;2-T.
[22] M. Truszcznyński, Generalized local colorings of graphs, J. Combin Theory (B) 54 (1992) 178-188, doi: 10.1016/0095-8956(92)90049-4.
[23] M. Truszcyński and Zs. Tuza, Linear upper bounds for local Ramsey numbers, Graphs and Combinatorics 3 (1987) 67-73.

Received 11 July 2000
Revised 3 September 2001