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Extremal problems of double stars

Ervin Győri, Runze Wang, Spencer Woolfson

Abstract

In a generalized Turán problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

Extremal problems of double stars

Abstract

In a generalized Turán problem, two graphs and are given and the question is the maximum number of copies of in an -free graph of order . In this paper, we study the number of double stars in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

Paper Structure

This paper contains 6 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Double Stars
  3. Adjacent Vertex Condition
  4. Non-adjacent vertices condition
  5. Number of Edges
  6. Number of Triangles

Key Result

Lemma 1

A triangle-free graph $G$ with $n$ vertices and maximum degree $\Delta$ contains at most $\Delta(n-\Delta)$ edges, and equality holds if and only if $\Delta \geq \frac{n}{2}$ and $G$ is the complete bipartite graph $K_{\Delta,n-\Delta}$. In particular, the balanced complete bipartite graph $K_{\lcei

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 14 more