Abstract
Let \(K\) be a set of \(k\) positive integers. A biclique cover of type \(K\) of a graph \(G\) is a collection of complete bipartite subgraphs of \(G\) such that for every edge \(e\) of \(G\), the number of bicliques need to cover \(e\) is a member of \(K\). If \(K=\{1,2,\ldots , k\}\) then the maximum number of vertices of a complete graph that admits a biclique cover of type \(K\) with \(d\) bicliques, \(n(k,d)\), is the maximum possible cardinality of a \(k\)-neighborly family of standard boxes in \(\mathbb {R}^d\). In this paper, we obtain an upper bound for \(n(k,d)\). Also, we show that the upper bound can be improved in some special cases. Moreover, we show that the existence of biclique cover of type \(K\) of the complete bipartite graph with a perfect matching removed is equivalent to the existence of a cross \(K\)-intersection family.
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Acknowledgments
The authors wish to express their thanks to Professor H. Hajiabolhassan for the useful conversations during the preparation of the paper. Also, the authors gratefully acknowledge the anonymous referees for their useful comments.
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This research was in part supported by a grant from research institute for ICT and is a part of Farokhlagha Moazami’s Ph.D. Thesis.
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Moazami, F., Soltankhah, N. On the Biclique Cover of the Complete Graph. Graphs and Combinatorics 31, 2347–2356 (2015). https://doi.org/10.1007/s00373-014-1493-2
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DOI: https://doi.org/10.1007/s00373-014-1493-2