Interval edge-colorings of complete graphs and -dimensional cubes

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