Non-r-partite graphs without complete split subgraphs

TL;DR

This paper characterizes the unique spectral extremal graphs in non-r-partite graphs avoiding certain complete split subgraphs, extending classical extremal and spectral graph theory results.

Contribution

It determines the unique spectral extremal graph for non-r-partite graphs avoiding a complete split graph, and shows spectral and size extremal graphs coincide for large n.

Findings

Identifies the unique spectral extremal graph in the specified class.

Proves the spectral extremal graphs are also size extremal for large n.

Extends classical extremal results to spectral extremal problems for specific non-r-partite graphs.

Abstract

The classical Simonovits' chromatic critical edge theorem shows that for sufficiently large $n$, if $H$ is an edge-color-critical graph with $\chi(H)=p+1\ge 3$, then the Tur\'an graph $T_{n,p}$ is the unique extremal graph with respect to ${\rm ex}(n,H)$. Denote by ${\rm EX}_{r+1}(n,H)$ and ${\rm SPEX}_{r+1}(n,H)$ the family of $n$-vertex $H$-free non-$r$-partite graphs with the maximum size and with the spectral radius, respectively. Li and Peng [SIAM J. Discrete Math. 37 (2023) 2462--2485] characterized the unique graph in $\mathrm{SPEX}_{r+1}(n,K_{r+1})$ for $r\geq 2$ and showed that $\mathrm{SPEX}_{r+1}(n,K_{r+1})\subseteq \mathrm{EX}_{r+1}(n,K_{r+1})$. It is interesting to study the extremal or spectral extremal problems for color-critical graph $H$ in non-$r$-partite graphs. For $p\geq 2$ and $q\geq 1$, we call the graph $B_{p,q}:=K_p\nabla qK_1$ a complete split graph (or generalized book graph). In this note, we determine the unique spectral extremal graph in $\mathrm{SPEX}_{r+1}(n,B_{p,q})$ and show that $\mathrm{SPEX}_{r+1}(n,B_{p,q})\subseteq \mathrm{EX}_{r+1}(n,B_{p,q})$ for sufficiently large $n$.