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# Pure and Applied Mathematics Quarterly

## Volume 18 (2022)

### Number 6

### Special issue in honor of Fan Chung

Guest editors: Paul Horn, Yong Lin, and Linyuan Lu

### Singular Turán numbers of stars

Pages: 2599 – 2618

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n6.a12

#### Authors

#### Abstract

Suppose that $G$ is a graph and $H$ is a subgraph of $G$. We call $H$ singular if the vertices of $H$ either have the same degree in $G$ or have pairwise distinct degrees in $G$. Let $T_S (n, H)$ be the largest number of edges of a graph with $n$ vertices that does not contain a singular copy of $H$. The problem of determining $T_S (n, H)$ was studied initially by Caro and Tuza, who obtained an asymptotic bound for each $H$. In this paper, we consider the case that $H$ is a star, and determine the exact values of $T_S (n, K_{1,2})$ for all $n$, $T_S (n, K_{1,4})$ and $T_S (n, K_{1,2s+1})$ for sufficiently large $n$.

#### Keywords

singular, Turán number, star, $H$-free

#### 2010 Mathematics Subject Classification

05C07, 05C35

Received 30 June 2021

Received revised 1 March 2022

Accepted 3 May 2022

Published 29 March 2023