Spectral supersaturation for color-critical graphs

TL;DR

This paper establishes spectral supersaturation results for color-critical graphs, linking spectral radius conditions to the number of subgraph copies and extremal graph structures, advancing extremal graph theory understanding.

Contribution

It provides the first complete spectral supersaturation results for color-critical graphs, resolving a problem by Ning-Zhai and extending prior extremal results.

Findings

Spectral radius thresholds guarantee many copies of color-critical graphs.

Extremal graphs with high spectral radius are close to Turán graphs.

The results are tight up to constant factors for certain parameters.

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 Bollob\'as-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Tur\'an graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $\lambda(G)$ 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 $\delta_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq \delta_F \sqrt{n}$, any $n$-vertex graph $G$ with $\lambda(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$ 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 $\lambda(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$, valid for all $1 \leq q \leq \delta_F n$. 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.