Long cycles in graphs through fragments

TL;DR

This paper establishes new lower bounds on the circumference of graphs based on connectivity, minimum degree, and fragments, completing the reverse versions of classical Dirac-type Hamiltonian conditions.

Contribution

It proves the reverse bounds for graph circumference related to Dirac-type conditions and introduces results involving fragments to support these bounds.

Findings

Proves lower bounds on graph circumference based on connectivity and degree.

Introduces results involving fragments to establish circumference bounds.

Completes the list of reverse Dirac-type conditions for graph cycles.

Abstract

Four basic Dirac-type sufficient conditions for a graph $G$ to be hamiltonian are known involving order $n$, minimum degree $\delta$, connectivity $\kappa$ and independence number $\alpha$ of $G$: (1) $\delta \geq n/2$ (Dirac); (2) $\kappa \geq 2$ and $\delta \geq (n+\kappa)/3$ (by the author); (3) $\kappa \geq 2$ and $\delta \geq \max\lbrace (n+2)/3,\alpha \rbrace$ (Nash-Williams); (4) $\kappa \geq 3$ and $\delta \geq \max\lbrace (n+2\kappa)/4,\alpha \rbrace$ (by the author). In this paper we prove the reverse version of (4) concerning the circumference $c$ of $G$ and completing the list of reverse versions of (1)-(4): (R1) if $\kappa \geq 2$, then $c\geq\min\lbrace n,2\delta\rbrace$ (Dirac); (R2) if $\kappa \geq 3$, then $c\geq\min\lbrace n,3\delta -\kappa\rbrace$ (by the author); (R3) if $\kappa\geq 3$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,3\delta-3\rbrace$ (Voss and Zuluaga); (R4) if $\kappa\geq 4$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,4\delta-2\kappa\rbrace$. To prove (R4), we present four more general results centered around a lower bound $c\geq 4\delta-2\kappa$ under four alternative conditions in terms of fragments. A subset $X$ of $V(G)$ is called a fragment of $G$ if $N(X)$ is a minimum cut-set and $V(G)-(X\cup N(X))\neq\emptyset$.