On the $4$-clique cover number of graphs

Abstract

In 1966, Erd\H{o}s, Goodman, and P\'osa proved that $\lfloor n^2/4 \rfloor$ cliques are sufficient to cover all edges in any $n$-vertex graph, with tightness achieved by the balanced complete bipartite graph. This result was generalized by Dau, Milenkovic, and Puleo, who showed that at most $\lfloor \frac n 3 \rfloor \lfloor \frac {n+1} 3 \rfloor \lfloor \frac {n+2} 3 \rfloor$ cliques are needed to cover all triangles in any $n$-vertex graph $G$, and the bound is best possible as witnessed by the balanced complete tripartite graph. They further conjectured that for $t \geq 4$, the $t$-clique cover number is maximized by the Tur\'an graph $T_{n,t}$. We confirm their conjecture for $t=4$ using novel techniques, including inductive frameworks, greedy partition method, local adjustments, and clique-counting lemmas by Erd\H{o}s and by Moon and Moser.