A Fan-type condition involving bipartite independence number for hamiltonicity in graphs

TL;DR

This paper establishes new conditions involving the bipartite independence number that guarantee Hamiltonicity and Hamiltonian-connectedness in 2-connected and 3-connected graphs, respectively.

Contribution

It introduces a fan-type condition based on the bipartite independence number for ensuring Hamiltonian properties in graphs, generalizing previous results.

Findings

Graphs satisfying the condition are Hamiltonian.

Graphs satisfying the stronger condition are Hamiltonian-connected.

The results extend recent work by Li and Liu.

Abstract

The bipartite independence number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s + t = q + 1$, such that for any two disjoint subsets $A, B \subseteq V(G)$ with $|A| = s$ and $|B| = t$, there exists an edge between $A$ and $B$. In this paper, we prove that for a 2-connected graph $G$ of order at least three, if $\max\{d_G(x), d_G(y)\} \ge \widetilde{\alpha}(G)$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian. Moreover, we prove that if $G$ is 3-connected and $\max\{d_G(x), d_G(y)\} \ge \widetilde{\alpha}(G)+1$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian-connected. Our results generalize the recent work by Li and Liu.