A refinement of theorems on vertex-disjoint chorded cycles

In 1963, Corr'adi and Hajnal settled a conjecture of Erd\H{o}s by proving that, for all $k\ge 1$, any graph $G$ with $\mathrm{\mid }G\mathrm{\mid }\ge 3k$ and minimum degree at least $2k$ contains $k$ vertex-disjoint cycles. In 2008, Finkel proved that for all $k\ge 1$, any graph $G$ with $\mathrm{\mid }G\mathrm{\mid }\ge 4k$ 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 $k\ge 1$, any graph $G$ with $\mathrm{\mid }G\mathrm{\mid }\ge 4k$ and minimum Ore-degree at least $6k-1$ contains $k$ vertex-disjoint cycles. We refine this result, characterizing the graphs $G$ with $\mathrm{\mid }G\mathrm{\mid }\ge 4k$ and minimum Ore-degree at least $6k-2$ that do not have $k$ disjoint chorded cycles.