Colorability in Line Graphs

Abstract

Chromatic graph theory is one of the most studied areas of graph theory, and this chapter looks at some of the problems that have line graph connections. A basic relationship is the fact that coloring the edges of a graph is equivalent to coloring the vertices of its line graph. In fact, this fact provides us with a line graph version of the four color theorem. The line graphs of cubic graphs constitute an interesting family on their own: it is known that their chromatic number is either 3 or 4, but which is a hard problem, and aspects of this are discussed here. We then proceed on to the general line graph problem with theorems that state that the line graph of a graph G with maximum degree Δ has chromatic number either Δ or Δ + 1 and it is always Δ if G is bipartite. The chapter concludes with modifications of the standard coloring requirement, one stronger and one weaker, and also results on coloring line graphs of multigraphs.