Edge Coloring of Bipartite Graphs with Constraints

Abstract

It is a classical result from graph theory that the edges of an l-regular bipartite graph can be colored using exactly l colors so that edges that share an endpoint are assigned different colors. In this paper we study two constrained versions of the bipartite edge coloring problem.

• Some of the edges adjacent to a pair of opposite vertices of an l-regular bipartite graph are already colored with S colors that appear only on one edge (single colors) and D colors that appear in two edges (double colors). We show that the rest of the edges can be colored using at most $\max \left\{ {\min \left\{ {l + D,\tfrac{{3l}} {2}} \right\},l + \tfrac{{S + D}} {2}} \right\}$ total colors. We also show that this bound is tight by constructing instances in which $\max \left\{ {\min \left\{ {l + D,\tfrac{{3l}} {2}} \right\},l + \tfrac{{S + D}} {2}} \right\}$ colors are indeed necessary.

• Some of the edges of an l-regular bipartite graph are already colored with S colors that appear only on one edge. We show that the rest of the edges can be colored using at most max{l + S/2, S} total colors. We also show that this bound is tight by constructing instances in which max{l + S/2, S total colors are necessary.

This work was partially funded by the Italian Ministry of University and Scientific Research under Project MURST 40% and ESPRIT LTR Project 20244 ALCOM-IT.