An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs

Abstract: We study the supersaturation problem in its edge-spectral form. Let $λ(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $λ(G)\le \sqrt{(1!-!1/r )2m}$. Recently, Li, Liu and Zhang proved the same bound for every $F$-free graph $G$, where $F$ is any color-critical graph with $χ(F)=r+1\ge4$, with equality only for regular complete $r$-partite graphs. It is then natural to ask how many copies of $F$ are forced once $λ(G)$ exceeds this threshold. Fang, Lin and Zhai answered this at the threshold itself, and conjectured that for any fixed $C&gt;0$, the condition $λ(G)\ge \sqrt{(1!-!{1}/{r})2m} +C$ forces $Ω!\left(m<sup>{(f-1)/2}\right)$ copies. In this paper, we answer this question with the best possible constant, proving that for every color-critical graph $F$ with $χ(F)=r+1\ge4$, there exists $δ_F&gt;0$ such that if $m$ is sufficiently large, $0&lt;q\leδ_F\sqrt m$, and $G$ is an $m$-edge graph with $λ^2(G)\ge 2\left(1-\tfrac1r\right)m+q$, then [ N_F(G)\ge\bigl(B_F-o(1)\bigr)\,q\, m^{{(f-2)}/{2}}, \quad \text{where}~~ B_F:=\tfrac{α_F}{4} (\tfrac{2r}{r-1} )^{{f}/{2}}, ] and the constant $B_F$ is best possible. Our result can be viewed as an edge-spectral counterpart of Mubayi's theorem, since it converts the spectral surplus $q$ into a linear number of copies with a sharp constant, and it solves the conjecture of Fang, Lin and Zhai in a stronger form.

• The paper proves that a spectral surplus q above Nikiforov’s threshold forces at least (B_F−o(1))q m^{(f−2)/2} copies of every color-critical graph F, with the best possible constant B_F.
• Its proof combines gap-preserving regularization, spectral stability, sharp local counting, and a first-order conversion between additive spectral gaps and internal edges.
• The result resolves the Fang–Lin–Zhai conjecture, while showing that matching-based constructions are optimal only for small gaps and that fixed-gap supersaturation can be nonlinear when r≥3.

Background and problem

The supersaturation problem asks how many copies of a fixed graph $F$ are forced once a host graph exceeds the Turán threshold. For color-critical graphs $F$ with $\chi(F)=r+1\geq 3$, Mubayi proved that if an $n$-vertex graph has $e(T_{n,r})+q$ edges with $1\leq q\leq \delta_F n$, then it contains at least $q\cdot c(n,F)$ copies of $F$, where $c(n,F)$ is the minimum number of copies created by adding one edge inside a part of the Turán graph $T_{n,r}$; Mubayi's estimate gives $F$0 for $F$1 (2607.01073).

The paper studies the edge-spectral analogue, in which the host graph is specified by its number of edges $F$2 rather than its order, and the threshold is expressed through the adjacency spectral radius $F$3. Nikiforov's theorem states that every $F$4-free graph satisfies $F$5, and Li–Liu–Zhang extended this to all color-critical $F$6 with $F$7. Fang, Lin and Zhai then proved that any $F$8-edge graph with $F$9 contains at least $\chi(F)=r+1\geq 3$0 copies of $\chi(F)=r+1\geq 3$1, unless $\chi(F)=r+1\geq 3$2 is regular complete $\chi(F)=r+1\geq 3$3-partite, and conjectured that exceeding the threshold by a fixed additive constant forces $\chi(F)=r+1\geq 3$4 copies.

Main results

The paper resolves this conjecture in a stronger form with sharp constants. The central result is: for every color-critical $\chi(F)=r+1\geq 3$5 of order $\chi(F)=r+1\geq 3$6 with $\chi(F)=r+1\geq 3$7, there exists $\chi(F)=r+1\geq 3$8 such that if $\chi(F)=r+1\geq 3$9 is large, $n$0, and $n$1, then

$n$2

and $n$3 is best possible. This is the exact edge-spectral counterpart of Mubayi's theorem: each unit of spectral surplus $n$4 converts linearly into $n$5 copies, just as each extra edge forces $n$6 copies in the classical setting. The extremal construction is the Turán graph with a matching added inside one part — adding a matching is the least efficient way to lift the spectral radius above the threshold.

The key technical theorem concerns a fixed additive gap in $n$7: for every $n$8 there exists $n$9 such that if $e(T_{n,r})+q$0 with $e(T_{n,r})+q$1, then

$e(T_{n,r})+q$2

and this coefficient is sharp: for arbitrarily large $e(T_{n,r})+q$3 there exist graphs meeting the spectral condition with at most $e(T_{n,r})+q$4 copies. Consequently,

$e(T_{n,r})+q$5

Since the result holds for all small $e(T_{n,r})+q$6, it immediately implies the Fang–Lin–Zhai conjecture for every fixed $e(T_{n,r})+q$7 (larger gaps only increase the count). The squared-gap theorem follows from the linear-gap theorem via the identity $e(T_{n,r})+q$8 for $e(T_{n,r})+q$9, where $1\leq q\leq \delta_F n$0; this conversion is valid precisely because $1\leq q\leq \delta_F n$1.

For cliques, Theorem applies with $1\leq q\leq \delta_F n$2, sharpening the earlier $1\leq q\leq \delta_F n$3 bound of Li–Liu–Zhang for $1\leq q\leq \delta_F n$4 into a statement with the optimal constant.

Method of proof

The lower bound proceeds by contradiction under the assumption $1\leq q\leq \delta_F n$5, which implies $1\leq q\leq \delta_F n$6. Four steps are involved:

Gap-preserving regularization. Iteratively deleting $1\leq q\leq \delta_F n$7-light edges (edges $1\leq q\leq \delta_F n$8 with $1\leq q\leq \delta_F n$9 for the unit Perron vector $q\cdot c(n,F)$0) and $q\cdot c(n,F)$1-deficient vertices never decreases the edge-spectral density $q\cdot c(n,F)$2. A counting argument against the edge-spectral supersaturation theorem shows the process stops after fewer than $q\cdot c(n,F)$3 deletions, yielding a subgraph $q\cdot c(n,F)$4 with $q\cdot c(n,F)$5 and $q\cdot c(n,F)$6 — almost all of the additive gap survives.

Stability and refinement. Since $q\cdot c(n,F)$7, edge-spectral stability provides a partition close to a balanced Turán graph. A hierarchy of constants ($q\cdot c(n,F)$8) drives a structural refinement showing the exceptional sets of low-degree and high-internal-degree vertices are empty, so every vertex has internal degree at most $q\cdot c(n,F)$9, few missing cross-edges, and an almost uniform Perron vector, $F$0.

Sharp local count. Each class-edge $F$1 (an edge inside some part) creates at least $F$2 copies of $F$3 whose only within-part edge is $F$4; these families are disjoint over distinct class-edges. This is a first-order sharpening of the per-edge count used at the threshold.

Gap conversion. Removing the $F$5 class-edges leaves an $F$6-partite graph, whose spectral radius is at most $F$7 by Nikiforov's theorem. Expanding $F$8 and using the near-uniformity of the Perron vector yields $F$9. Summing the local counts gives $c(n,F)$0, contradicting the assumption.

Three ingredients are new relative to Fang–Lin–Zhai: the regularization preserving the additive gap, the first-order-accurate per-edge count, and the sharp conversion $c(n,F)$1 from spectral gap to class-edge count.

Sharpness and the nonlinear regime

Tightness is witnessed by $c(n,F)$2, the balanced Turán graph with a matching of size $c(n,F)$3 added inside one part. An equitable-partition computation gives $c(n,F)$4, while $c(n,F)$5, since embeddings using two or more matching edges contribute only $c(n,F)$6 copies. Choosing $c(n,F)$7 establishes the upper bound matching the lower bound to first order.

A notable caveat is that linearity holds only as $c(n,F)$8. If instead a star with $c(n,F)$9 edges is added inside one part, the spectral lift is $T_{n,r}$0, so a fixed gap $T_{n,r}$1 is achieved with $T_{n,r}$2 internal edges and hence strictly fewer copies. Thus for fixed $T_{n,r}$3 the true minimum is nonlinear in $T_{n,r}$4, and the matching construction is optimal only to first order. This contrasts with the triangle case ($T_{n,r}$5, $T_{n,r}$6), where Chen–Li–Tang showed the forced count is exactly linear in the gap for all $T_{n,r}$7, with split graphs extremal; the nonlinearity is a new feature of the case $T_{n,r}$8.

Limitations and open questions

The main theorem covers the range $T_{n,r}$9 of spectral surplus; beyond this range, once $F$00 the count jumps to $F$01 by the Erdős–Simonovits supersaturation applied along the stability partition, but the intermediate range $F$02 is not understood. The paper also leaves open the pointwise function

$F$03

known only to satisfy $F$04 for small $F$05; determining $F$06 exactly — equivalently, identifying the optimal "internal" graph added to a part of $F$07 — remains open, as does deciding for which $F$08 the function $F$09 is linear. Further directions include the case $F$10, where the spectral extremal graphs are split graphs rather than Turán graphs and require a different stability analysis (odd cycles $F$11, $F$12, being a concrete target); general (non-color-critical) forbidden graphs; bipartite $F$13 with additive spectral gaps above split graphs; analogues for the $F$14-spectral and signless Laplacian radii; and removing the $F$15 error to obtain exact counts in the spirit of Lovász–Simonovits and Liu–Pikhurko–Staden.

Conclusion

This paper completes the edge-spectral line of supersaturation results for color-critical graphs: it converts an additive spectral surplus $F$16 above Nikiforov's threshold into a linear number of copies of $F$17 with the best possible constant $F$18, thereby solving the Fang–Lin–Zhai conjecture in a strengthened form and providing the precise edge-spectral analogue of Mubayi's theorem. The proof introduces a reusable mechanism — gap-preserving regularization combined with a first-order conversion of spectral gap into internal-edge count — and the sharpness analysis reveals a genuinely nonlinear structure for fixed gaps when $F$19, distinguishing the color-critical case from the classical triangle setting.