Interval edge-colorings of complete graphs and nn-dimensional cubes

An edge-coloring of a graph GG with colors 1,2,…,t1,2,…,t is called an interval t-coloring   if for each i∈{1,2,…,t}i∈{1,2,…,t} there is at least one edge of GG colored by ii, and the colors of edges incident to any vertex of GG are distinct and form an interval of integers. In this paper we show that if n=p2qn=p2q, where pp is odd, qq is nonnegative, and 2n−1≤t≤4n−2−p−q2n−1≤t≤4n−2−p−q, then the complete graph K2nK2n has an interval tt-coloring. We also prove that if n≤t≤n(n+1)2, then the nn-dimensional cube QnQn has an interval tt-coloring.