# On degree sum conditions for 2-factors with a prescribed number of cycles

## TL;DR

This paper establishes new degree sum conditions in highly connected graphs that guarantee the existence of a 2-factor with a specified number of cycles, generalizing previous results in the field.

## Contribution

It introduces a generalized degree condition involving maximum degree sums of independent sets that ensures the existence of 2-factors with a given number of cycles, extending prior work.

## Key findings

- Proves a degree sum condition for 2-factors with exactly k cycles
- Generalizes previous results by Brandt et al. and Yamashita
- Applicable to m-connected graphs of order n ≥ 5k - 2

## Abstract

For a vertex subset $X$ of a graph $G$, let $\Delta_{t}(X)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n \ge 5k - 2$. If $\Delta_{2}(X) \ge n$ for every independent set $X$ of size $\lceil m/k \rceil+1$ in $G$, then $G$ has a 2-factor with exactly $k$ cycles. This is a common generalization of the results obtained by Brandt et al. [Degree conditions for 2-factors, J. Graph Theory 24 (1997) 165-173] and Yamashita [On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math. 308 (2008) 6584-6587], respectively.