The Abu-Khzam$\unicode{x2013}$Langston Conjecture for Graphs with $α(G) = 2$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911723775066112 |
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| author | Dahlke, Jonathan C. |
| author_facet | Dahlke, Jonathan C. |
| contents | 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 $K_{χ(G)}$. 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 $Δ(G)$ colours. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28159 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Abu-Khzam$\unicode{x2013}$Langston Conjecture for Graphs with $α(G) = 2$ Dahlke, Jonathan C. Combinatorics 05C83, 05C15 (Primary) 05C70 (Secondary) 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 $K_{χ(G)}$. 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 $Δ(G)$ colours. |
| title | The Abu-Khzam$\unicode{x2013}$Langston Conjecture for Graphs with $α(G) = 2$ |
| topic | Combinatorics 05C83, 05C15 (Primary) 05C70 (Secondary) |
| url | https://arxiv.org/abs/2605.28159 |