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

TL;DR

This paper establishes degree conditions under which a graph can be partitioned into a specified number of triangles and quadrilaterals, providing partial proof for El-Zahar's Conjecture.

Contribution

It introduces new degree sum conditions that guarantee the existence of a partition into triangles and quadrilaterals, advancing understanding of graph decompositions.

Findings

Graphs satisfying the degree sum condition contain the specified partitions.

Partial proof of El-Zahar's Conjecture for certain parameters.

Conditions are optimal for the given partition types.

Abstract

For two positive integers $r$ and $s$ with $r\geq 2s-2$, if $G$ is a graph of order $3r+4s$ such that $d(x)+d(y)\geq 4r+4s$ for every $xy\not\in E(G)$, then $G$ independently contains $r$ triangles and $s$ quadrilaterals, which partially prove the El-Zahar's Conjecture.