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arXiv · 2605.28159

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

Jonathan C. Dahlke
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Abstract

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 $\Delta(G)$ colours.

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Jonathan C. Dahlke. 2026-05-27. The Abu-Khzam$\unicode{x2013}$Langston Conjecture for Graphs with $\alpha(G) = 2$. https://arxiv.org/abs/2605.28159

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