A Study on Degree Characterization for Hamiltonian-Connected Graphs

Abstract

Consider any undirected and simple graph \(G = (V, E)\), where \(V\) and \(E\) denote the vertex set and the edge set of \(G\), respectively. Let \(|G| = |V| = n \geq 3\). The well-known Ore’s theorem states that if \(\deg_G(u) + \deg_G(v) \geq n + k\) holds for each pair of nonadjacent vertices \(u\) and \(v\) of \(G\), then \(G\) is traceable for \(k = -1\), Hamiltonian for \(k = 0\), and Hamiltonian-connected for \(k = 1\). In this paper, we investigate any graph \(G\) with \(\deg_G(u) + \deg_G(v) \geq n – 1\) for any nonadjacent vertex pair \(\{u,v\}\) of \(G\), in particular. We call it the \((*)\) condition. We derive four graph families, \(\mathcal{H}_i\), \(1 \leq i \leq 4\), and prove that all graphs satisfying \((*)\) are Hamiltonian-connected unless \(G \in \bigcup_{i=1}^{4} \mathcal{H}_i\). We also establish a comprehensive theorem for \(G\) satisfying \((*)\), which shows that \(G\) is traceable, Hamiltonian, pancyclic, or Hamiltonian-connected unless \(G\) belongs to different subsets of \(\{\mathcal{H}_i | 1 \leq i \leq 4\}\), respectively.