Partitioning a graph into cycles with a specified number of chords

TL;DR

This paper proves that large graphs with high degree sum can be partitioned into a specified number of cycles, each containing a minimum number of chords, extending previous results in graph theory.

Contribution

It establishes a new condition under which a graph can be partitioned into cycles with a given number of chords, generalizing earlier theorems for large graphs.

Findings

Graphs with high degree sum can be partitioned into cycles with many chords.

The result applies to sufficiently large graphs, depending on parameters.

It extends previous work by providing a bound for the number of chords in each cycle.

Abstract

For a graph $G$, let $\sigma_{2}(G)$ be the minimum degree sum of two non-adjacent vertices in $G$. A chord of a cycle in a graph $G$ is an edge of $G$ joining two non-consecutive vertices of the cycle. In this paper, we prove the following result, which is an extension of a result of Brandt et al. (J. Graph Theory 24 (1997) 165-173) for large graphs: For positive integers $k$ and $c$, there exists an integer $f(k,c)$ such that, if $G$ is a graph of order $n \ge f(k, c)$ and $\sigma_{2}(G) \ge n$, then $G$ can be partitioned into $k$ vertex-disjoint cycles, each of which has at least $c$ chords.