Factoring complete graphs and hypergraphs into factors with few maximal cliques

TL;DR

This paper studies how to decompose complete hypergraphs into factors with minimal total number of maximal cliques, extending known results from graphs to hypergraphs and multiple factors.

Contribution

It generalizes the concept of minimal clique sum factorizations from graphs to hypergraphs and multiple factors, providing new bounds and characterizations.

Findings

Determines bounds for $f_r(t,n)$ when $r>2$ or $t>2$.

Characterizes graphs with $c(G)+c(ar{G})=n+2$.

Extends classical graph results to hypergraph factorizations.

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 $c(G)+c(\overline G)=n+2$.