Graphs and Combinatorics (1997) 13: 325 -343 Degree Sum Conditions for Hamiltonicity on k-Partite Graphs Graphs and Combinatorics © Springer-Verlag 1997 Guantao Chen!" and Michael S. Jacobson"! 1 Department of Mathematics and Computer Science, Georgia State University, Atlanta, GA 30303, USA. e-mail: matgcc@gsusgi2.gsu.edu 2 Department of Mathematics, University of Louisville, Louisville, KY 40292, USA. e-mail: msjacoOl @homer.louisville.edu Abstract. One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graph G has order p and minimum degree at least then G is hamiltonian. Moon and Moser showed that if G is a balanced bipartite graph (the two partite sets have the same order) with minimum degree more than then G is hamiltonian. Recently their idea is generalized to k-partite graphs by Chen, Faudree, Gould, Jacobson, and Lesniak in terms of minimum degrees. In this paper, we generalize this result in terms of degree sum and the following result is obtained: Let G be a balanced k-partite graph with order kn. If for every pair of nonadjacent vertices u and v which are in different parts then G is hamiltonian. 1. Introduction {( k__ 2 )n k + 1 d(u) + d(v) > ( k__ 4 )n k+2 if k is odd if k is even One of the earliest results in the theory of hamiltonian graphs is due to Dirac [1]. Theorem 1 (Dirac). If G is a graph with p 3 vertices having minimum degree c5(G) then G is hamiltonian. In 1960, Ore [10] generalized the above results by using degree sums. • Research supported by N.S.A. grant # MDA 904-95-1-1091 Research supported by O.N.R. grant # NOOOI4-J-91-1098