Edge-spectral Turán theorems for color-critical graphs with applications
Yongtao Li, Hong Liu, Shengtong Zhang
TL;DR
The paper develops an edge-spectral framework for Turán-type extremal problems, proving that for color-critical graphs F with χ(F)=r+1≥4 and sufficiently large m, any F-free graph G with m edges satisfies λ^2(G) ≤ (1-1/r)2m, with equality only for regular complete r-partite graphs. It introduces an edge-spectral stability method, together with Perron–Frobenius vector stability, to force near-structured partitions and derive exact extremals, resolving open problems (notably by Yu and Li) and extending prior spectral results. For almost-bipartite forbidden graphs, the authors obtain an asymptotic formula λ(G) = √m + max_{M∈𝔐_F} e(M)/v(M) + O(m^{-1/2}) and describe an A∪C structure with MF-membership constraints, enabling sharp determinations of extremal graphs for several classical substructures. The work also provides broad applications to planar and k-planar graphs, theta and wheel-related families, and classical bipartite graphs, delivering unified, sharp results and confirming multiple conjectures, while introducing new spectral techniques for stability of Perron–Frobenius vectors. The findings deepen the connection between edge-density constraints and spectral radius, offering a powerful tool for future Brualdi–Hoffman–Turán-type problems.
Abstract
A classical result of Nosal asserts that every $m$-edge graph with spectral radius $λ(G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [35] states that if $G$ is an $m$-edge graph with $λ(G)> \sqrt{(1- {1}/{r})2m}$, then $G$ contains a clique $K_{r+1}$. This result implies the Turán 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 $χ(F)=r+1\ge 4$, if $m$ is sufficiently large and $G$ is an $F$-free graph with $m$ edges, then $λ(G)\le \sqrt{(1- {1}/{r})2m}$, 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.