Interval colorings of complete balanced multipartite graphs

A graph G is called a complete k-partite (k ≥ 2) graph if its vertices can be partitioned into k independent sets V1, . . . , Vk such that each vertex in Vi is adjacent to all the other vertices in Vj for 1 ≤ i < j ≤ k. A complete k-partite graph G is a complete balanced kpartite graph if |V1| = |V2| = · · · = |Vk|. An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if G has an interval t-coloring for some positive integer t. In this paper we show that a complete balanced k-partite graph G with n vertices in each part is interval colorable if and only if nk is even. We also prove that if nk is even and (k−1)n ≤ t ≤ (