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

Summary: Some results on interval edge colorings of
(, )-biregular bipartite graphs
A.S. Asratian and C. J. Casselgren
Department of Mathematics, Link¨oping University
S-581 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 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 (, )-biregular bipartite graph G has an interval t-coloring is
NP-complete 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 t-coloring then + -gcd(, ) t m-1, 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 t-coloring for every


Source: Asratian, Armen - Matematiska Institutionen, Linköpings Universitet


Collections: Mathematics