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New type degree conditions for a graph to have a 2-factor

Masaki Kashima

TL;DR

This work introduces new degree-sum–independent-set conditions that guarantee a 2-factor in graphs. By defining $σ_k(G)$ and the local degree parameter $δ_G(I)$, the authors prove two relaxations of classical Hamiltonicity criteria: (i) if every independent set satisfies $|I|\le δ_G(I)-1$, then $G$ has a 2-factor, and (ii) if every independent set satisfies $|I|\le δ_G(I)$, then $G$ has a 2-factor or belongs to the exceptional class $\mathcal{G}$, which do not possess a 2-factor. The proofs fuse a key local improvement lemma, Tutte's 2-factor theorem, and barrier analysis to establish existence results and to characterize the exceptional cases. The paper then extends the framework to 2-factors with a bounded number of components, proving that under a strengthened degree-sum condition and a suitable cut-set structure, a 2-factor with at most $k$ cycles exists; a corollary further connects these conditions to 1-tough graphs and dominating cycles. Overall, the results broaden the toolkit for 2-factor existence, link degree-sum and independence structure to cycle packing, and clarify how near-Dirac/Chvátal-Erdős conditions govern factor structure.

Abstract

A 2-factor of a graph is a 2-regular spanning subgraph. For a graph $G$ and an independent set $I$ of $G$, let $δ_G(I)$ denote the minimum degree of vertices contained in $I$. We show that (1) if every independent set $I$ of $G$ satisfies $|I|\leq δ_G(I)-1$, then $G$ has a 2-factor and that (2) if every independent set $I$ of $G$ satisfies $|I|\leq δ_G(I)$, then $G$ has a 2-factor unless $G$ is isomorphic to a graph in completely determined exceptional graphs. It can be easily shown that the assumption of (1) is a relaxation of the Dirac condition on Hamiltonicity of graphs, and that the assumption of (2) is a relaxation of the Chvátal-Erdős condition on Hamiltonicity of graphs. Furthermore, for graphs with the assumption of (1), we show some results on a 2-factor with a bounded number of cycles.

New type degree conditions for a graph to have a 2-factor

TL;DR

This work introduces new degree-sum–independent-set conditions that guarantee a 2-factor in graphs. By defining and the local degree parameter , the authors prove two relaxations of classical Hamiltonicity criteria: (i) if every independent set satisfies , then has a 2-factor, and (ii) if every independent set satisfies , then has a 2-factor or belongs to the exceptional class , which do not possess a 2-factor. The proofs fuse a key local improvement lemma, Tutte's 2-factor theorem, and barrier analysis to establish existence results and to characterize the exceptional cases. The paper then extends the framework to 2-factors with a bounded number of components, proving that under a strengthened degree-sum condition and a suitable cut-set structure, a 2-factor with at most cycles exists; a corollary further connects these conditions to 1-tough graphs and dominating cycles. Overall, the results broaden the toolkit for 2-factor existence, link degree-sum and independence structure to cycle packing, and clarify how near-Dirac/Chvátal-Erdős conditions govern factor structure.

Abstract

A 2-factor of a graph is a 2-regular spanning subgraph. For a graph and an independent set of , let denote the minimum degree of vertices contained in . We show that (1) if every independent set of satisfies , then has a 2-factor and that (2) if every independent set of satisfies , then has a 2-factor unless is isomorphic to a graph in completely determined exceptional graphs. It can be easily shown that the assumption of (1) is a relaxation of the Dirac condition on Hamiltonicity of graphs, and that the assumption of (2) is a relaxation of the Chvátal-Erdős condition on Hamiltonicity of graphs. Furthermore, for graphs with the assumption of (1), we show some results on a 2-factor with a bounded number of cycles.

Paper Structure

This paper contains 10 sections, 12 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. New type degree conditions for a graph to have a 2-factor
  3. 2-factors with bounded number of cycles
  4. Existence of a 2-factor
  5. Proof of Theorem \ref{['thm:sigma0']}
  6. Tutte's 2-factor theorem
  7. Proof of Theorem \ref{['thm:sigma1']}
  8. 2-factor with a bounded number of components
  9. Proof of Theorem \ref{['thm:sigma0 k comp']}
  10. Proof of Corollary \ref{['cor:sigma0 2 comp']}

Key Result

Theorem 1

Let $G$ be a graph of order $n$ and let $k$ be a positive integer. If $n\geq 4k$ and $\sigma_2(G)\geq n$, then $G$ has a 2-factor with exactly $k$ cycles.

Theorems & Definitions (21)

  • Theorem 1: BCFGL1997
  • Theorem 2: C2018
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Conjecture 7
  • Theorem 8
  • ...and 11 more