# On k-intersection edge colourings

• Volume: 29, Issue: 2, page 411-418
• ISSN: 2083-5892

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## Abstract

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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 $fₖ\left(\Delta \right)=ma{x}_{G:\Delta \left(G\right)=\Delta }{\chi }^{\text{'}}ₖ\left(G\right)$. We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

## How to cite

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Rahul Muthu, N. Narayanan, and C.R. Subramanian. "On k-intersection edge colourings." Discussiones Mathematicae Graph Theory 29.2 (2009): 411-418. <http://eudml.org/doc/270823>.

@article{RahulMuthu2009,
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 $fₖ(Δ) = max_\{G : Δ(G) = Δ\} \{χ^\{\prime \}ₖ(G)\}$. We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.},
author = {Rahul Muthu, N. Narayanan, C.R. Subramanian},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph theory; k-intersection edge colouring; probabilistic method; -intersection edge colouring},
language = {eng},
number = {2},
pages = {411-418},
title = {On k-intersection edge colourings},
url = {http://eudml.org/doc/270823},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Rahul Muthu
AU - N. Narayanan
AU - C.R. Subramanian
TI - On k-intersection edge colourings
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 2
SP - 411
EP - 418
AB - 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 $fₖ(Δ) = max_{G : Δ(G) = Δ} {χ^{\prime }ₖ(G)}$. We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.
LA - eng
KW - graph theory; k-intersection edge colouring; probabilistic method; -intersection edge colouring
UR - http://eudml.org/doc/270823
ER -

## References

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1. [1] N. Alon and B. Mohar, Chromatic number of graph powers, Combinatorics Probability and Computing 11 (2002) 1-10, doi: 10.1017/S0963548301004965. Zbl0991.05042
2. [2] N. Alon and J. Spencer, The Probabilistic Method (John Wiley, 2000). Zbl0996.05001
3. [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 Zbl0886.05068
4. [4] P. Erdös and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in: Infinite and Finite Sets, 1975.
5. [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. Zbl0507.65027
6. [6] M. Molloy and B. Reed, Graph Coloring and the Probabilistic Method (Springer, Algorithms and Combinatorics, 2002).
7. [7] R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge University Press, 1995). Zbl0849.68039
8. [8] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Metody Diskret. Analys. (1964) 25-30.

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