Hamiltonicity in balancedk-partite graphs

Abstract

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least$\frac{p}{2}$ thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than$\frac{p}{4}$ thenG is hamiltonian. In this paper, their idea is generalized tok-partite graphs and the following result is obtained: LetG be a balancedk-partite graph with orderp = kn. If the minimum degree

$$\delta (G) > \left\{ {\begin{array}{*{20}c} {\left( {\frac{k}{2} - \frac{1}{{k + 1}}} \right)n if k is odd } \\ {\left( {\frac{k}{2} - \frac{2}{{k + 2}}} \right)n if k is even} \\ \end{array} } \right.$$

thenG is hamiltonian. The result is best possible.