A degree sum condition on the order, the connectivity and the independence number for Hamiltonicity

TL;DR

This paper proves a conjecture linking degree sums, connectivity, and independence number to Hamiltonicity in graphs, improving previous bounds and establishing the condition's optimality.

Contribution

It settles a conjecture by proving a new degree sum condition that guarantees Hamiltonian cycles in k-connected graphs, extending and strengthening prior results.

Findings

Proves the conjecture on degree sum conditions for Hamiltonicity.

Improves previous bounds on degree sum conditions.

Establishes the optimality of the degree sum condition.

Abstract

In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+\kappa(G)+(k-2)(\alpha(G)-1)$, then $G$ contains a Hamiltonian cycle, where $\sigma_{k+1}(G)$, $\kappa(G)$ and $\alpha(G)$ are the minimum degree sum of $k+1$ independent vertices, the connectivity and the independence number of $G$, respectively. In this paper, we settle this conjecture. This is an improvement of the result obtained by Li: If $G$ is a $k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+(k-1)(\alpha(G)-1)$, then $G$ is Hamiltonian. The degree sum condition is best possible.