Hamiltonian Properties of the \(3-(\gamma,2)\)-Critical Graphs

Abstract

Ewa Wojcicka (Journal of Graph Theory, \(14(1990), 205-215)\) showed that every connected, 3-color-critical graph on more than 6 vertices has a Hamiltonian path. Henning et al. (Discrete Mathematics, \(161(1996), 175-184)\) defined a graph \(G\) to be \(k\)-\((\gamma, d)\)-critical graph if \(\gamma(G) = k\) and \(\gamma(G + uv) = k – 1\) for each pair \(u, v\) of nonadjacent vertices of \(G\) that are at distance at most \(d\) apart. They asked if a 3-\((\gamma, 2)\)-critical graph must contain a dominating path. In this paper, we show that every connected, 3-\((\gamma, 2)\)-critical graph must contain a dominating path. Further, we show that every connected, 3-\((\gamma, 2)\)-critical graph on more than 6 vertices has a Hamiltonian path.