Discussiones Mathematicae Graph Theory 29(2) (2009) 411-418
doi: 10.7151/dmgt.1456

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Rahul Muthu, N. Narayanan and C.R. Subramanian

The Institute of Mathematical Sciences
Taramani, Chennai-600113, India
e-mail: {rahulm,narayan,crs}@imsc.res.in


We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ′k(G). Let fk be defined by
fk(Δ) =
G : Δ(G) = Δ 

We show that fk(Δ) = Θ([(Δ2)/k]). We also discuss some open problems.

Keywords: graph theory, k-intersection edge colouring, probabilistic method.

2000 Mathematics Subject Classification: 05C15, 05D40.


[1] N. Alon and B. Mohar, Chromatic number of graph powers, Combinatorics Probability and Computing 11 (2002) 1-10, doi: 10.1017/S0963548301004965.
[2] N. Alon and J. Spencer, The Probabilistic Method (John Wiley, 2000).
[3] A.C. Burris and R.H. Schelp, Vertex-distinguishing proper edge colourings, J. Graph Theory 26 (1997) 70-82, doi: 10.1002/(SICI)1097-0118(199710)26:2<73::AID-JGT2>3.0.CO;2-C.
[4] P. Erdös and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in: Infinite and Finite Sets, 1975.
[5] S.T. McCormick, Optimal approximation of sparse Hessians and its equivalence to a graph colouring problem, Mathematical Programming 26 (1983) 153-171, doi: 10.1007/BF02592052.
[6] M. Molloy and B. Reed, Graph Coloring and the Probabilistic Method (Springer, Algorithms and Combinatorics, 2002).
[7] R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge University Press, 1995).
[8] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Metody Diskret. Analys. (1964) 25-30.

Received 3 December 2007
Revised 14 February 2009
Accepted 14 February 2009