Disjoint chorded cycles in a $2$-connected graph

Abstract

A chorded cycle in a graph $G$ is a cycle containing an edge of $G$ that joins two nonconsecutive vertices of the cycle. In 2010, Gao and Qiao independently proved that 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. Moreover, the neighborhood-union bound $4s$ is sharp.

26 pages,3 figures