Edge-spectral Tur\'{a}n theorems for color-critical graphs with applications

TL;DR

This paper establishes spectral bounds for graphs avoiding certain subgraphs, especially color-critical and almost-bipartite graphs, using edge-spectral methods, and characterizes extremal structures for large graphs.

Contribution

It proves new spectral Turán-type theorems for color-critical and almost-bipartite graphs, resolving open problems and conjectures in spectral extremal graph theory.

Findings

Spectral bounds for color-critical graphs are established.

Structural characterization of extremal graphs avoiding almost-bipartite graphs.

Resolution of open problems and conjectures in spectral extremal graph theory.

Abstract

A classical result of Nosal asserts that every $m$-edge graph with spectral radius $\lambda (G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [35] states that if $G$ is an $m$-edge graph with $\lambda (G)> \sqrt{(1- {1}/{r})2m}$, then $G$ contains a clique $K_{r+1}$. This result implies the Tur\'{a}n theorem and Wilf theorem, and offers a new perspective on the existence of substructures. The edge-spectral conditions are versatile for enforcing substructures, as they can be applied to any sparse graph regardless of its edge density. In this paper, we prove that for any color-critical graph $F$ with chromatic number $\chi (F)=r+1\ge 4$, if $m$ is sufficiently large and $G$ is an $F$-free graph with $m$ edges, then $\lambda (G)\le \sqrt{(1- {1}/{r})2m}$, with equality if and only if $G$ is a regular complete $r$-partite graph. This settles an open problem proposed by Yu and Li [52] and also gives spectral bounds for graphs forbidding books and wheels. Secondly, we establish an asymptotic formula and structural characterization when we forbid an almost-bipartite graph $F$, where $F$ is called almost-bipartite if it can be made bipartite by removing at most one edge. As applications, we determine the unique $m$-edge spectral extremal graph for every integer $m$ when avoiding certain substructures, including complete bipartite graphs plus an edge, cycles plus an edge, and theta graphs, etc. Our results resolve an open problem proposed by Li, Zhao and Zou [24], as well as two conjectures posed by Liu and Li [31]. The arguments in our proofs are based on the edge-spectral stability method recently established by the authors. In addition, we develop some new spectral techniques, including the stability result for the Perron--Frobenius eigenvector.