On Degree Sum Conditions and Vertex-Disjoint Chorded Cycles

TL;DR

This paper establishes a sharp degree sum condition ensuring the existence of multiple vertex-disjoint chorded cycles in large graphs, advancing understanding of cycle structures under degree constraints.

Contribution

The paper introduces a new degree sum condition involving $\sigma_t(G)$ that guarantees multiple vertex-disjoint chorded cycles, extending previous cycle existence results.

Findings

Proves that $\sigma_t(G) extgreater= 3kt - t + 1$ implies $k$ vertex-disjoint chorded cycles.

Shows the degree sum condition is sharp by constructing extremal graphs.

Provides insights into graphs lacking chorded cycles under similar degree conditions.

Abstract

In this paper, we consider a general degree sum condition sufficient to imply the existence of $k$ vertex-disjoint chorded cycles in a graph $G$. Let $\sigma_t(G)$ be the minimum degree sum of $t$ independent vertices of $G$. We prove that if $G$ is a graph of sufficiently large order and $\sigma_t(G)\geq 3kt-t+1$ with $k\geq 1$, then $G$ contains $k$ vertex-disjoint chorded cycles. We also show that the degree sum condition on $\sigma_t(G)$ is sharp. To do this, we also investigate graphs without chorded cycles.