## ON k-INTERSECTION EDGE COLOURINGS

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

## Abstract

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) = max G : Δ(G) = Δ {χ′k(G)}.

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.

## References

 [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.