Hadwiger’s conjecture for line graphs

We prove that Hadwiger’s conjecture holds for line graphs. Equivalently, we show that for every loopless graph G (possibly with parallel edges) and every integer k≥0, either G is k-edge-colourable, or there are k+1 connected subgraphs A1,…,Ak+1 of G, each with at least one edge, such that E(Ai∩Aj)=∅ and V(Ai∩Aj)≠∅ for 1≤i<j≤k.