Clique covers and decompositions of cliques of graphs

TL;DR

This paper investigates optimal clique cover and decomposition problems in graphs, proving fractional and asymptotic results for conjectures related to covering edges and larger cliques, using advanced combinatorial techniques.

Contribution

It provides fractional relaxations and asymptotically optimal solutions for conjectures on clique covers and decompositions, introducing a new framework with Zykov symmetrization, the Frankl-R"odl nibble, and Szemerédi's Regularity Lemma.

Findings

Proved fractional relaxations of Erdős's conjecture on clique decompositions.

Established asymptotically optimal versions of conjectures on covering larger cliques.

Developed a new framework combining symmetrization, nibble method, and regularity lemma.

Abstract

In 1966, Erd\H{o}s, Goodman, and P\'{o}sa showed that if $G$ is an $n$-vertex graph, then at most $\lfloor n^2/4 \rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced complete bipartite graph. This was generalized independently by Gy\H{o}ri--Kostochka, Kahn, and Chung, who showed that every $n$-vertex graph admits an edge-decomposition into cliques of total `cost' at most $2 \lfloor n^2/4 \rfloor$, where an $i$-vertex clique has cost $i$. Erd\H{o}s suggested the following strengthening: every $n$-vertex graph admits an edge-decomposition into cliques of total cost at most $\lfloor n^2/4 \rfloor$, where now an $i$-vertex clique has cost $i-1$. We prove fractional relaxations and asymptotically optimal versions of both this conjecture and a conjecture of Dau, Milenkovic, and Puleo on covering the $t$-vertex cliques of a graph instead of the edges. Our proofs introduce a general framework for these problems using Zykov symmetrization, the Frankl-R\"odl nibble method, and the Szemer\'edi Regularity Lemma.