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Coloring Graphs With No Totally Odd Clique Immersion
[Submitted on 11 Aug 2025 (v1), last revised 2 Jul 2026 (this ve · 2025-08-11 · via math updates on arXiv.org

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Abstract:We prove that graphs that do not contain a totally odd immersion of $K_t$ are $\mathcal{O}(t)$-colorable. In particular, we show that any graph with no totally odd immersion of $K_t$ is the union of a bipartite graph and a graph which forbids an immersion of $K_{\mathcal{O}(t)}$. Our results are algorithmic, and we give a fixed-parameter tractable algorithm (in $t$) to find such a decomposition.

Submission history

From: Caleb McFarland [view email]
[v1] Mon, 11 Aug 2025 15:57:58 UTC (22 KB)
[v2] Tue, 19 Aug 2025 21:00:41 UTC (22 KB)
[v3] Thu, 2 Jul 2026 14:14:35 UTC (25 KB)