Sharp bounds for decomposing graphs into edges and triangles

Abstract

In 1966, Erdos, Goodman and Posa proved that the edges of an n-vertex graph can be decomposed into at most n^2/4 cliques. Moreover, such a decomposition may consist of edges and triangles only. In 1980s, Gyori and Kostochka and independently Chung strengthened the first result in order to answer a question of Katona and Tarjan by proving that the minimum sum of the clique-orders in such decompositions is at most n^2/2.

In 1987, these results led Gyori and Tuza to conjecture that the edge-set of an n-vertex graph can be decomposed into m edges and t triangles with 2m+3t <= n^2/2 + O(1). Recently, Kral, Lidicky, Martins and Pehova proved the conjecture asymptotically, i.e., found an edge-decomposition of any n-vertex graph into m edges and t triangles with 2m+3t<=n^2/2 + o(n^2). In this work, we fully resolve the conjecture of Gyori and Tuza. Specifically, we prove the only large enough n-vertex graphs that cannot be decomposed into m edges and t triangles with 2m+3t <= n^2/2 are n-vertex cliques with n congruent to 4 (mod 6).