The Abu-Khzam$\text{unicode}x2013$Langston Conjecture for Graphs with $\alpha (G)=2$

The Abu-Khzam–Langston conjecture, that is the weak-immersion analogue of Hadwiger’s conjecture and a weak version of an earlier conjecture of Lescure and Meyniel, asserts that every graph $G$ contains a weak immersion of $K_{\chi (G)}$. We prove the conjecture for the class of graphs of independence number two. Along the way, we introduce a notion of \emph{cycle-matching colouring} of a graph, a relaxation of edge-colouring in which colour classes induce vertex-disjoint unions of edges and odd cycles, and prove a sharpening of Vizing’s theorem in this setting: every multigraph admits a cycle-matching colouring with at most $\mathrm{\Delta }(G)$ colours.