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

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.

💡 Research Summary

The paper addresses a classic problem in extremal graph theory: under what degree conditions does a graph contain a prescribed collection of vertex‑disjoint cycles? Specifically, it focuses on the case where the desired cycles are triangles (C₃) and quadrilaterals (C₄). The authors consider two positive integers r and s with the additional restriction r ≥ 2s − 2 and study graphs G of order n = 3r + 4s. The main hypothesis is a degree‑sum condition for non‑adjacent vertex pairs, namely σ₂(G) = min{d(x)+d(y) | xy∉E(G)} ≥ n + r, which is equivalent to d(x)+d(y) ≥ 4r + 4s for every non‑edge xy.

The paper builds on earlier work. Aigner and Brandt proved that a minimum‑degree condition δ(G) ≥ ⌈3r/2⌉ + ⌈4s/2⌉ guarantees r disjoint triangles and s disjoint quadrilaterals. Brandt et al. and later Yan showed that the weaker σ₂(G) ≥ n + r condition yields r triangles and s cycles of length at most four, but only s − 1 of those cycles can be guaranteed to be quadrilaterals; the remaining four vertices form an unspecified subgraph D with at least four edges.

The authors first re‑establish Yan’s result (Lemma 3.1) that under σ₂(G) ≥ n + r the graph contains r vertex‑disjoint C₃’s and s − 1 vertex‑disjoint C₄’s. The novelty lies in Theorem 3.2, where they prove that the leftover four‑vertex subgraph D can be forced to have at least four edges and, after a careful restructuring, to contain a C₄. The proof proceeds by repeatedly examining the connections between D and the already selected triangles and quadrilaterals (collectively denoted H). If D lacks two independent edges, the authors replace a cycle in H by a larger cycle that uses two vertices of D, thereby increasing the edge count inside D. By exploiting Lemmas 2.2–2.5 (which guarantee the existence of a C₄ or a C₃ together with a path of length three when enough edges join a small path or a small cycle to another structure), they eventually ensure that D contains a C₄ or can be transformed into a configuration that does.

The final step, Theorem 3.3, uses the extra numerical condition r ≥ 2s − 2. Assuming D is not already a C₄, the authors show that D must be isomorphic to a specific four‑vertex graph F₄ (a claw with an extra edge). They then analyze the number of edges between the vertex set U of F₄ and the collection of already placed triangles and quadrilaterals. If any quadrilateral in H receives nine or more edges from U, Lemma 2.7 yields two disjoint C₄’s, contradicting the assumption that D lacks a C₄. If any triangle receives seven or more edges from U, Lemma 2.9 gives a C₃ and a C₄, again a contradiction. By bounding the total number of edges from U to H using the condition r ≥ 2s − 2, they derive a contradiction unless D already contains a C₄. Consequently, G must contain r disjoint triangles and s disjoint quadrilaterals.

Thus the paper partially confirms El‑Zahar’s conjecture for the special case where the desired cycle lengths are only 3 or 4, under a degree‑sum condition that is stronger than a simple minimum‑degree bound. The result improves upon earlier theorems by eliminating the “missing” quadrilateral and showing that the extra structural condition r ≥ 2s − 2 is sufficient to guarantee a full partition into the prescribed cycles. The techniques—particularly the systematic use of edge‑count arguments between a small residual subgraph and the already constructed cycles—provide a useful toolkit for future investigations of cycle packings under degree constraints.