Totally odd strong immersion conjecture

Prove that every graph G contains the complete graph K_{χ(G)} as a totally odd strong immersion.

Background

The totally odd strong immersion conjecture strengthens the Lescure–Meyniel immersion conjecture in a different direction from Hajós’ conjecture. It requires the edges of the immersed complete graph to be represented by odd paths and additionally requires the immersion to be strong. The paper investigates whether minimal counterexamples to this conjecture can arise through standard graph products, proving that they cannot arise from the Cartesian, lexicographic, direct, or strong product.

References

Therefore, the following conjecture (which is inspired by one of Churchley ) extends Conjecture~\ref{conj:imm} to arbitrarily dense graph classes.

— Totally odd immersions of complete graphs in graph products  (2502.10227 - Echeverría et al., 14 Feb 2025) in Section 1.1, Conjecture 2 (Conjecture \ref{conj:church})

Therefore, the following conjecture (which is inspired by one of Churchley ) extends Conjecture~\ref{conj:imm} to arbitrarily dense graph classes. Every graph $G$ contains $K_{\chi(G)}$ as a totally odd strong immersion.

— Totally odd immersions of complete graphs in graph products  (2502.10227 - Echeverría et al., 14 Feb 2025) in Section 1, Subsection “Totally odd immersions”; Conjecture 2 (labelled \ref{conj:church})