 
Summary: Some results on interval edge colorings of
(, )biregular bipartite graphs
A.S. Asratian and C. J. Casselgren
Department of Mathematics, Link¨oping University
S581 83 Link¨oping, Sweden
Abstract. A bipartite graph G is called (, )biregular if all vertices in
one part of G have the degree and all vertices in the other part have the
degree . An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called
an interval tcoloring 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 (, )biregular bipartite graph G has an interval tcoloring is
NPcomplete in the case when > 3 and is a divisor of . It is
known that if an (, )biregular bipartite graph G on m vertices has an
interval tcoloring then + gcd(, ) t m1, where gcd(, ) is the
greatest common divisor of and . We prove that if an (, )biregular
bipartite graph has m 2( + ) 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 (, )biregular bipartite graph G
which has m = n( + ) vertices and admits an interval tcoloring for every
