An Ore-type condition for hamiltonicity in tough graphs and the extremal examples

Masahiro Sanka ; Songling Shan¹

¹Auburn University

The electronic journal of combinatorics, Tome 31 (2024) no. 1

Résumé

Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $t$ is between 1 and 2, and recently the second author proved a general result. The result states that if the degree sum of any two nonadjacent vertices of $G$ is greater than $\frac{2n}{t+1}+t-2$, then $G$ is hamiltonian. It was conjectured in the same paper that the "$+t$" in the bound $\frac{2n}{t+1}+t-2$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $t$-tough graphs $G$ on $n\ge 3$ vertices for which $\sigma_2(G) = \frac{2n}{t+1}-2$ but $G$ is non-hamiltonian.

Détail

DOI : 10.37236/12483

Classification : 05C45, 05C38
Mots-clés : Ore's theorem, toughness of a graph, Chvátal's toughness conjecture