A strengthening of the spectral chromatic critical edge theorem: books and theta graphs

TL;DR

This paper strengthens spectral versions of the chromatic critical edge theorem by identifying new classes of graphs, books and theta graphs, for which spectral radius conditions guarantee the presence of certain subgraphs.

Contribution

It introduces spectral thresholds for books and theta graphs, expanding the classes of graphs where spectral conditions imply structural properties.

Findings

Graphs with spectral radius greater than that of $T_{n,2}$ contain large books.

Graphs with spectral radius greater than that of $T_{n,2}$ contain cycles of length up to $n/7.

Results relate to longstanding conjectures and questions in spectral graph theory.

Abstract

The chromatic critical edge theorem of Simonovits states that for a given color critical graph $H$ with $\chi(H)=k+1$, there exists an $n_0(H)$ such that the Tur\'an graph $T_{n,k}$ is the only extremal graph with respect to $ex(n,H)$ provided $n \geq n_0(H)$. Nikiforov's pioneer work on spectral graph theory implies that the color critical edge theorem also holds if $ex(n,H)$ is replaced by the maximum spectral radius and $n_0(H)$ is an exponential function of $|H|$. We want to know which color critical graphs $H$ satisfy that $n_0(H)$ is a linear function of $|H|$. Previous graphs include complete graphs and odd cycles. In this paper, we find two new classes of graphs: books and theta graphs. Namely, we prove that every graph on $n$ vertices with $\rho(G)>\rho(T_{n,2})$ contains a book of size greater than $\frac{n}{6.5}$. This can be seen as a spectral version of a 1962 conjecture by Erd\H{o}s, which states that every graph on $n$ vertices with $e(G)>e(T_{n,2})$ contains a book of size greater than $\frac{n}{6}$. In addition, our result on theta graphs implies that if $G$ is a graph of order $n$ with $\rho(G)>\rho(T_{n,2})$, then $G$ contains a cycle of length $t$ for every $t\leq \frac{n}{7}$. This is related to an open question by Nikiforov which asks to determine the maximum $c$ such that every graph $G$ of large enough order $n$ with $\rho(G)>\rho(T_{n,2})$ contains a cycle of length $t$ for every $t\leq cn$.