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arXiv · 2309.03083

Factoring complete graphs and hypergraphs into factors with few maximal cliques

Paul Erdős·David P. Galvin·Fred Galvin·Michael M. Krieger
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Abstract

For integers $r,t\geq2$ and $n\geq1$ let $f_r(t,n)$ be the minimum, over all factorizations of the complete $r$-uniform hypergraph of order $n$ into $t$ factors $H_1,\dots,H_t$, of $\sum_{i=1}^tc(H_i)$ where $c(H_i)$ is the number of maximal cliques in $H_i$. It is known that $f_2(2,n)=n+1$; in fact, if $G$ is a graph of order $n$, then $c(G)+c(\overline G)\geq n+1$ with equality iff $\omega(G)+\alpha(G)=n+1$ where $\omega$ is the clique number and $\alpha$ the independence number. In this paper we investigate $f_r(t,n)$ when $r>2$ or $t>2$. We also characterize graphs $G$ of order $n$ with $c(G)+c(\overline G)=n+2$.

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Paul Erdős, David P. Galvin, Fred Galvin, Michael M. Krieger. 2023-09-01. Factoring complete graphs and hypergraphs into factors with few maximal cliques. https://arxiv.org/abs/2309.03083

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