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 χint′(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 χint′(G,X) in terms of the vertex degrees of G. We also determine χint′(G,X) exactly for some classes of bipartite graphs G. Furthermore, we present upper bounds on χint′(G,X) for classes of bipartite graphs G with maximum degree Δ(G) at most 9: in particular, if Δ(G)=4, then χint′(G,X)≤6; if Δ(G)=5, then χint′(G,X)≤15; if Δ(G)=6, then χint′(G,X)≤33.