Some results on interval edge colorings of (α, β)-biregular bipartite graphs

A bipartite graph G is called (α, β)-biregular if all vertices in one part of G have the degree α and all vertices in the other part have the degree β. An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored i, for i = 1, . . . , t. We show that the problem to determine whether an (α, β)-biregular bipartite graph G has an interval t-coloring is NP-complete in the case when α > β ≥ 3 and β is a divisor of α. It is known that if an (α, β)-biregular bipartite graph G on m vertices has an interval t-coloring then α+β−gcd(α, β) ≤ t ≤ m−1, where gcd(α, β) is the greatest common divisor of α and β. We prove that if an (α, β)-biregular bipartite graph has m ≥ 2(α + β) vertices then the upper bound can be improved to m− 3. We also show that this bound is tight by constructing, for every integer n ≥ 1, a connected (α, β)-biregular bipartite graph G which has m = n(α+ β) vertices and admits an interval t-coloring for every t satisfying α+ β − gcd(α, β) ≤ t ≤ m− 3.