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Degree powers and number of stars in graphs with a forbidden broom

Dániel Gerbner

Abstract

Given a graph $G$ with degree sequence $d_1,\dots, d_n$ and a positive integer $r$, let $e_r(G)=\sum_{i=1}^n d_i^r$. We denote by $\mathrm{ex}_r(n,F)$ the largest value of $e_r(G)$ among $n$-vertex $F$-free graphs $G$, and by $\mathrm{ex}(n,S_r,G)$ the largest number of stars $S_r$ in $n$-vertex $F$-free graphs. The \textit{broom} $B(\ell,s)$ is the graph obtained from an $\ell$-vertex path by adding $s$ new leaves connected to a penultimate vertex $v$ of the path. We determine $\mathrm{ex}_r(n,B(\ell,s))$ for $r\ge 2$, any $\ell,s$ and sufficiently large $n$, proving a conjecture of Lan, Liu, Qin and Shi. We also determine $\mathrm{ex}(n,S_r,B(\ell,s))$ for $r\ge 2$, any $\ell,s$ and sufficiently large $n$.

Degree powers and number of stars in graphs with a forbidden broom

Abstract

Given a graph with degree sequence and a positive integer , let . We denote by the largest value of among -vertex -free graphs , and by the largest number of stars in -vertex -free graphs. The \textit{broom} is the graph obtained from an -vertex path by adding new leaves connected to a penultimate vertex of the path. We determine for , any and sufficiently large , proving a conjecture of Lan, Liu, Qin and Shi. We also determine for , any and sufficiently large .
Paper Structure (2 sections, 1 theorem)

This paper contains 2 sections, 1 theorem.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

(i) If $\ell$ is even, $r\ge 2$ and $n$ is sufficiently large, then $\mathrm{ex}_r(n,B(\ell,s))=e_r(H(k,n))$. If $\ell$ is odd and $\ell\ge 7$ or $\ell=5$ and $s=0$, then $\mathrm{ex}_r(n,B(\ell,s))=e_r(H^*(k,n))$. If $\ell=5$ and $s>0$, then $\mathrm{ex}_r(n,B(\ell,s))=e_r(F_n)$. Moreover, $H(k,n)$

Theorems & Definitions (11)

  • Theorem 1.1
  • Claim 2.1
  • proof : Proof of Claim
  • Claim 2.2
  • proof : Proof of Claim
  • Claim 2.3
  • proof : Proof of Claim
  • Claim 2.4
  • proof : Proof of Claim
  • Claim 2.5
  • ...and 1 more