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arXiv · 1709.01590

On the Triangle Clique Cover and $K_t$ Clique Cover Problems

Hoang Dau·Olgica Milenkovic·Gregory J. Puleo
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Abstract

An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t \geq 1$. In particular, we extend a classical result of Erd\"os, Goodman, and P\'osa (1966) on the edge clique cover number ($t = 2$), also known as the intersection number, to the case $t = 3$. The upper bound is tight, with equality holding only for the Tur\'an graph $T(n,3)$. We also extend an algorithm of Scheinerman and Trenk (1999) to solve a weighted version of the $K_t$ clique cover problem on a superclass of chordal graphs. We also prove that the $K_t$ clique cover problem is NP-hard.

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Hoang Dau, Olgica Milenkovic, Gregory J. Puleo. 2017-09-05. On the Triangle Clique Cover and $K_t$ Clique Cover Problems. https://doi.org/10.1016/j.disc.2019.111627

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