Spectral supersaturation for color-critical graphs
Longfei Fang, Yongtao Li, Huiqiu Lin, Jie Ma
TL;DR
This work develops a complete spectral supersaturation theory for color-critical graphs $F$ with $\\chi(F)=r+1$, advancing beyond classical edge-count results by leveraging a spectral incremental framework around the Turán graph $T_{n,r}$. The authors prove two central theorems: (Y) a sqrt{n}-range result that, under a lower-bound spectral condition, yields at least $q c(n,F)$ copies of $F$ with extremals in $\\mathcal{T}_{n,r,q}$ (minimized by $Y_{n,r,q}$), and (Z) a linear-range companion showing that with a higher spectral threshold, either the extremal is $L_{n,r,q}$ and the copy count meets a near-optimal bound or the copies are bounded away from this threshold. The core methodology combines the graph removal lemma, spectral stability, and a pair of key lemmas (first-key, second-key) to control spectral radius and graph structure when deviating from Turán form. The results resolve the Ning–Zhai problem in full generality, unify spectral supersaturation with classical extremal supersaturation, and yield applications to $F$-covering and related conjectures (notably Li–Lu–Peng).
Abstract
A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential line of research initiated by Bollobás-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Turán graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $λ(G)$ be the spectral radius of a graph $G$. We first prove that for any color-critical graph $F$ with chromatic number $r+1$, there exists $δ_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq δ_F \sqrt{n}$, any $n$-vertex graph $G$ with $λ(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} λ(T)$ contains at least $q \cdot c(n,F)$ copies of $F$, where $c(n,F)$ denotes the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$; moreover, any extremal graph $G$ must belong to $ \mathcal{T}_{n,r,q}$.Next, we prove a spectral supersaturation result for the analogous condition $λ(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} λ(T)$, valid for all $1 \leq q \leq δ_F n$. Together, these results provide a complete resolution to a problem proposed by Ning-Zhai, and establish a spectral counterpart to the well-known results of Mubayi and Pikhurko-Yilma in the extremal supersaturation setting. A notable feature of our first result is that the restriction $q = O(\sqrt{n})$ is tight up to a constant factor, in contrast to the linear bounds provided by other settings discussed above. As applications, we extend a result of Liu-Mubayi, and solve a related conjecture by Li-Lu-Peng.