Lescure–Meyniel immersion conjecture

Prove that every graph G contains the complete graph K_{χ(G)} as an immersion.

Background

The Lescure–Meyniel conjecture weakens Hajós’ subdivision conjecture by replacing subdivisions with immersions. It asserts that the chromatic number of every graph determines the order of a complete graph immersion contained in that graph. The paper explicitly identifies this conjecture as unresolved and studies whether graph products can generate minimal counterexamples to it.

References

In 1989, Lescure and Meyniel made a conjecture that is a weakening of Hajós' and that remains open: every graph contains an immersion of $K_{\chi(G)}$.

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

The following conjecture of Lescure and Meyniel has received considerable attention in recent years . Every graph $G$ contains $K_{\chi(G)}$ as an immersion.

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