Asymmetric odd-cycle homomorphism threshold conjecture

Prove that every $C_{2t+3}$-free graph with minimum degree at least $delta|G|$ is homomorphic to a constant-size $C_{2t+1}$-free graph, for every integer $t\ge 1$ and every $delta>0$; equivalently, establish that $delta_{\mathrm{hom}}(C_{2t+3};C_{2t+1})=0$.

Background

The paper proves that a C2t+5C_{2t+5}-free graph with linear minimum degree admits a homomorphism to a constant-size C2t+1C_{2t+1}-free graph. The conjecture asks whether the excluded odd cycle can be shortened from C2t+5C_{2t+5} to C2t+3C_{2t+3} without changing the conclusion.

If true, the result would be best possible because the symmetric threshold satisfies δhom(C2t+1;C2t+1)=1/(2t+1)>0\delta_{\mathrm{hom}}(C_{2t+1};C_{2t+1})=1/(2t+1)>0. The paper establishes the conjecture under additional bounded-domination-number or bounded-VC-dimension assumptions.

References

In fact, we believe that the inclusion of $C_{2t+5}$ in \cref{thm:C3C5C7} is unnecessary, and leave the following as a tantalizing open problem.

— Asymmetric results about graph homomorphisms  (2502.20278 - Gishboliner et al., 27 Feb 2025) in Conjecture 1.4, Section 1, subsection “Asymmetric homomorphism thresholds”

In fact, we believe that the inclusion of $C_{2t+5}$ in \cref{thm:C3C5C7} is unnecessary, and leave the following as a tantalizing open problem. If $G$ is a $C_{2t+3}$-free graph with minimum degree at least $\delta\abs G$, then $G$ is homomorphic to a $C_{2t+1}$-free graph of constant size (depending only on $\delta$). In other words, $\delta_{\hom}(C_{2t+3};C_{2t+1})=0$.

— Asymmetric results about graph homomorphisms  (2502.20278 - Gishboliner et al., 27 Feb 2025) in Conjecture 1.6, Section 1.2 (labelled conj:no C7)