Linear vertex-cut bound for two longest cycles

Establish that there exists a constant C>0 such that, for every 2-connected graph G and any two longest cycles X and Y in G, a vertex set S of size at most C·|V(X)∩V(Y)| separates X and Y in G.

Background

The paper’s structural arguments prove a linear lower bound on the intersection of two longest cycles in highly connected graphs, but they do not establish the stronger local-separation conclusion proposed here.

The conjecture is motivated by earlier quantitative approaches, which showed that a small intersection forces a relatively small vertex cut separating the two cycles. Such a separation theorem would have consequences for lower bounds on the circumference of vertex-transitive graphs, including an Ω(n{2/3}) bound.

References

There exists a constant $C>0$ such that, for every $2$-connected graph $G$ and any two longest cycles $X$ and $Y$ in $G$, there is a set $S\subseteq V(G)$ with $|S|\leq C\cdot |V(X)\cap V(Y)|$ that separates $X$ and $Y$ in $G.

— Longest cycles intersect linearly in highly connected graphs  (2609.20724 - Ma et al., 17 Sep 2026) in Section 6.3, Conjecture 6