The Hamiltonian Number of Graphs with Prescribed Connectivity

Abstract

A Hamiltonian walk in a connected graph \(G\) is a closed walk of minimum length which contains every vertex of \(G\). The Hamiltonian number \(h(G)\) of a connected graph \(G\) is the length of a Hamiltonian walk in \(G\). Let \(\mathcal{G}(n)\) be the set of all connected graphs of order \(n\), \(\mathcal{G}(n, \kappa = k)\) be the set of all graphs in \(\mathcal{G}(n)\) having connectivity \(\kappa = k\), and \(h(n,k) = \{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\). We prove in this paper that for any pair of integers \(n\) and \(k\) with \(1 \leq k \leq n – 1\), there exist positive integers \(a := \min(h;n,k)) = \min\{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\) and \(b := \max(h;n,k)) = \max\{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\) such that \((h;n,k) = \{x \in \mathbb{Z} : a \leq x \leq b\}\). The values of \(\min(h;n,k))\) and \(\max(h(n,k))\) are obtained in all situations.