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On the Chromatic Number of Non-Sparse Random Intersection Graphs

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Abstract

An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random intersection graphs (RIGs) (as defined in Karoński et al. Comb. Probab. Comput. J. 8, 131–159 (1999); Singer-Cohen (1995)) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability p. Such graphs can capture interactions in networks due to sharing of resources among nodes. In this paper, we discuss various structural and algorithmic results concerning random intersection graphs and we focus on the computational problem of properly coloring random instances of the binomial random intersection graphs model. For the latter, we consider a range of parameters m, p for which RIGs differ substantially from Erdős-Rényi random graphs and for which greedy approaches fail.

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Notes

  1. Note however, that this does not mean that the chromatic number is close to np, since the part that is not coloured could be a clique in the worst case.

  2. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Consequently, the clique number of a perfect graph is equal to its chromatic number.

  3. More precisely, if \(\mathcal {B}\) is the set of different label choices that can give rise to a graph G, then the problem of inferring the complete information of label choices from G is solvable if there is some \(B^{*} \in \mathcal {B}\) such that Pr(B ∗|G) > Pr(B|G), for all \(\mathcal {B} \ni B \neq B^{*}\).

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Authors and Affiliations

  1. Computer Technology Institute and Press “Diophantus”, Patras, Greece

    Sotiris E. Nikoletseas, Christoforos L. Raptopoulos & Paul G. Spirakis

  2. Computer Science Department, University of Patras, Patras, Greece

    Sotiris E. Nikoletseas, Christoforos L. Raptopoulos & Paul G. Spirakis

  3. Department of Computer Science, University of Liverpool, Liverpool, UK

    Paul G. Spirakis

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  1. Sotiris E. Nikoletseas
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  2. Christoforos L. Raptopoulos
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  3. Paul G. Spirakis
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Correspondence to Christoforos L. Raptopoulos.

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This article is part of the Topical Collection on 50th Anniversary

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Nikoletseas, S.E., Raptopoulos, C.L. & Spirakis, P.G. On the Chromatic Number of Non-Sparse Random Intersection Graphs. Theory Comput Syst 60, 112–127 (2017). https://doi.org/10.1007/s00224-016-9733-x

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