One-sided interval edge-colorings of bipartite graphs

Abstract Let G be a bipartite graph with bipartition ( X , Y ) . An X -interval coloring of G is a proper edge-coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by χ i n t ′ ( G , X ) the minimum k such that G has an X -interval coloring with k colors. In this paper we give various upper and lower bounds on χ i n t ′ ( G , X ) in terms of the vertex degrees of G . We also determine χ i n t ′ ( G , X ) exactly for some classes of bipartite graphs G . Furthermore, we present upper bounds on χ i n t ′ ( G , X ) for classes of bipartite graphs G with maximum degree Δ ( G ) at most 9 : in particular, if Δ ( G ) = 4 , then χ i n t ′ ( G , X ) ≤ 6 ; if Δ ( G ) = 5 , then χ i n t ′ ( G , X ) ≤ 15 ; if Δ ( G ) = 6 , then χ i n t ′ ( G , X ) ≤ 33 .

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