Regarding two conjectures on clique and biclique partitions

For a graph \(G\), let \(cp(G)\) denote the minimum number of cliques of \(G\) needed to cover the edges of \(G\) exactly once. Similarly, let \(bp_k(G)\) denote the minimum number of bicliques (i.e. complete bipartite subgraphs of \(G\)) needed to cover each edge of \(G\) exactly \(k\) times. We co...

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Bibliographic Details
Published inarXiv.org
Main Authors Rohatgi, Dhruv, Urschel, John C, Wellens, Jake
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.05.2020
Subjects
Graph theory
Upper bounds
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Summary:For a graph \(G\), let \(cp(G)\) denote the minimum number of cliques of \(G\) needed to cover the edges of \(G\) exactly once. Similarly, let \(bp_k(G)\) denote the minimum number of bicliques (i.e. complete bipartite subgraphs of \(G\)) needed to cover each edge of \(G\) exactly \(k\) times. We consider two conjectures -- one regarding the maximum possible value of \(cp(G) + cp(\overline{G})\) (due to de Caen, Erdős, Pullman and Wormald) and the other regarding \(bp_k(K_n)\) (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on \(\max_G cp(G) + cp(\overline{G})\), and we prove an asymptotic version of the second, showing that \(bp_k(K_n) = (1+o(1))n\).
ISSN:2331-8422