Degree conditions for the partition of a graph into triangles and quadrilaterals

Abstract

For two positive integers $r$ and $s$ with $r\ge 2s-2$, if $G$ is a graph of order $3r+4s$ such that $d(x)+d(y)\ge 4r+4s$ for every $xy\notin E(G)$, then $G$ independently contains $r$ triangles and $s$ quadrilaterals, which partially prove the El-Zahar's Conjecture.