A refinement of theorems on vertex-disjoint chorded cycles

TL;DR

This paper refines existing theorems on the conditions under which graphs contain multiple vertex-disjoint chorded cycles, specifically characterizing graphs that nearly meet the previous bounds but lack such cycles.

Contribution

It provides a precise characterization of graphs with minimum Ore-degree at least 6k-2 that do not contain k vertex-disjoint chorded cycles, refining prior bounds.

Findings

Characterization of graphs with Ore-degree at least 6k-2 lacking k disjoint chorded cycles.

Extension of Finkel's and Chiba et al.'s results to tighter bounds.

Identification of extremal graph structures related to chorded cycle disjointness.

Abstract

In 1963, Corr\'adi and Hajnal settled a conjecture of Erd\H{o}s by proving that, for all $k \geq 1$, any graph $G$ with $|G| \geq 3k$ and minimum degree at least $2k$ contains $k$ vertex-disjoint cycles. In 2008, Finkel proved that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum degree at least $3k$ contains $k$ vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum Ore-degree at least $6k-1$ contains $k$ vertex-disjoint cycles. We refine this result, characterizing the graphs $G$ with $|G| \geq 4k$ and minimum Ore-degree at least $6k-2$ that do not have $k$ disjoint chorded cycles.