Cyclability, Connectivity and Circumference

TL;DR

This paper explores the relationships between cyclability, connectivity, and circumference in graphs, establishing bounds and conditions under which graphs are hamiltonian or have large cycles, with new generalizations and extremal results.

Contribution

It introduces new bounds linking cyclability and circumference, generalizes existing extremal results, and characterizes extremal graphs for hamiltonicity in highly connected graphs.

Findings

For k ≤ √(n+3), k-cyclable graphs have circumference ≥ 2k.

When k ≤ 3n/4 - 1, circumference ≥ k+2.

Graphs with certain edge counts are hamiltonian and extremal graphs are unique.

Abstract

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$ \textit{ordered} vertices are present in a common cycle in that order, then the graph is said to be $k$-ordered. We show that when $k \leq \sqrt{n+3}$, $k$-cyclable graphs also have circumference $c(G) \geq 2k$, and that this is best possible. Furthermore when $k \leq \frac{3n}{4} -1$, $c(G) \geq k+2$, and for $k$-ordered graphs we show $c(G) \geq \min\{n,2k\}$. We also generalize a result by Byer et al. on the maximum number of edges in nonhamiltonian $k$-connected graphs, and show that if $G$ is a $k$-connected graph of order $n \geq 2(k^2+k)$ with $|E(G)| > \binom{n-k}{2} + k^2$, then the graph is hamiltonian, and moreover the extremal graphs are unique.