Disjoint chorded cycles in a $2$-connected graph

TL;DR

The paper extends the existences of $s$ vertex-disjoint chorded cycles to $2$-connected graphs with order at least $4s$ under the neighborhood-union condition $ abla_2(G)\\ge 4s$ (or when $G$ is complete). The authors employ the framework of optimal systems of chorded cycles and a suite of technical lemmas to bound how the remainder of the graph interacts with the system, ultimately proving the theorem by contradiction. This work resolves Gould's question for the 2-connected case and tightens the understanding of when chorded cycles can be packed disjointly. The approaches provide a structured method for analyzing chorded-cycle packing via optimal systems and detailed component analysis, which could inform related packing problems in graphs.

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.

Figures (3)

• Figure 1: Exceptions for $|E_H(V_1,V_2)|=3$
• Figure 2: Exceptions for $|E_H(V_1,V_2)|=4$
• Figure 3: Chorded cycles in $H$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Lemma 2.7