A generalization of interval edge-colorings of graphs

An edge-coloring of a graph GG with colors 1,2,…,t1,2,…,t is called an interval (t,1)(t,1)-coloring if at least one edge of GG is colored by ii, i=1,2,…,ti=1,2,…,t, and the colors of edges incident to each vertex of GG are distinct and form an interval of integers with no more than one gap. In this paper we investigate some properties of interval (t,1)(t,1)-colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some families of graphs.