Disjoint chorded cycles in a $2$-connected graph

Abstract

A chorded cycle in a graph $G$ is a cycle on which two nonadjacent vertices are adjacent in the graph $G$. In 2010, Gao and Qiao independently proved a graph of order at least $4s$, in which the neighborhood union of any two nonadjacent vertices has at least $4s+1$ vertices, contains $s$ vertex-disjoint chorded cycles. In 2022, Gould raised a problem that asks whether increasing connectivity would improve the neighborhood union condition. In this paper, we solve the problem for $2$-connected graphs by proving that a $2$-connected graph of order at least $4s$, in which the neighborhood union of any two nonadjacent vertices has at least $4s$ vertices, contains $s$ vertex-disjoint chorded cycles.