An upper bound for the chromatic number of line graphs

It was conjectured by Reed [B. Reed, ω , α , and χ , Journal of Graph Theory 27 (1998) 177–212] that for any graph G , the graph’s chromatic number χ ( G ) is bounded above by ⌈ Δ ( G ) + 1 + ω ( G ) 2 ⌉ , where Δ ( G ) and ω ( G ) are the maximum degree and clique number of G , respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.