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arXiv · 1909.04873

Extremal graphs with local covering conditions

Debsoumya Chakraborti·Po-Shen Loh
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Abstract

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We completely solve this problem when $H$ is a clique, as well as more generally when $H$ is any regular graph with degree at least about half its number of vertices. We also characterize the extremal graphs when $H$ is an Erd\H{o}s-R\'enyi random graph. The extremal structures turn out to have the similar form as the conjectured extremal structures for a well-studied but elusive problem of similar flavor with local constraints: to maximize the number of copies of a fixed clique in graphs in which all degrees have a fixed upper bound.

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Debsoumya Chakraborti, Po-Shen Loh. 2019-09-11. Extremal graphs with local covering conditions. https://doi.org/10.1137/19m1286712

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