Establish spectral supersaturation for even cycles

Determine whether, for every integer \(t\geq3\), there exists a largest constant \(c(t)>0\) such that every sufficiently large \(n\)-vertex graph \(G\) satisfying \(\lambda(G)>\lambda(S_{t-1,n}^+)\), with \(S_{t-1,n}^+=K_{t-1}\vee I_{n-t+1}^+\), contains at least \(c(t)n^{t-3}\) copies of \(C_{2t}\), and determine the largest such constant if it exists.

Background

The paper explains that its framework applies to color-critical graphs but does not directly extend to degenerate bipartite graphs such as even cycles. For these graphs, the relevant extremal structures are sparse and neither the Turán graph nor the matching-based construction Yn,r,qY_{n,r,q} has an evident analogue.

A prior result cited in the paper shows that exceeding the spectral radius of St−1,n+S_{t-1,n}^+ forces at least one C2tC_{2t}. The proposed problem asks for a quantitative supersaturation bound of order nt−3n^{t-3}, together with the optimal leading constant.

References

For every integer $t \ge 3$, does there exist a constant $c(t) >0$ such that if $n$ is sufficiently large and $G$ is an $n$-vertex graph with $\lambda (G) > \lambda (S_{t-1,n}+)$, then $$ #\, C_{2t} (G) \ge c(t)\cdot n{t-3} \ ? $$ If the above is true, what is the largest value of such $c(t)$?

— Optimal spectral supersaturation for cliques and odd cycles  (2610.06038 - Chen et al., 5 Oct 2026) in Problem “Counting even cycles,” Section 6, Concluding remarks

For degenerate $F$, such as $C_{2t}$ or $K_{t,t}$, the Tur\'{a}n number is $o(n2)$ and the extremal graphs are sparse, so neither $T_{n,r}$ nor $Y_{n,r,q}$ has an obvious analogue, and even the correct order of the threshold $q$ is unclear.

— Optimal spectral supersaturation for cliques and odd cycles  (2610.06038 - Chen et al., 5 Oct 2026) in Section 6, Concluding remarks, subsection “Counting degenerate bipartite graphs”