Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi-inducibility of 4-vertex graphs

Levente Bodnár, Oleg Pikhurko

Abstract

For a graph $H$ whose edges are coloured blue or red, the $H$-semi-inducibility problem asks for the maximum, over all graphs $G$ of given order $n$, of the number of injections from the vertex set of $H$ into the vertex set of $G$ that send red (resp. blue) edges of $H$ to edges (resp. non-edges) of $G$. We consider all possible 4-vertex non-complete graphs $H$ and essentially resolve all remaining cases except when $H$ is the 3-edge path coloured blue-blue-red in this order (or is equivalent to this case). Some of our proofs are computer-generated, using the flag algebra method of Razborov.

Semi-inducibility of 4-vertex graphs

Abstract

For a graph whose edges are coloured blue or red, the -semi-inducibility problem asks for the maximum, over all graphs of given order , of the number of injections from the vertex set of into the vertex set of that send red (resp. blue) edges of to edges (resp. non-edges) of . We consider all possible 4-vertex non-complete graphs and essentially resolve all remaining cases except when is the 3-edge path coloured blue-blue-red in this order (or is equivalent to this case). Some of our proofs are computer-generated, using the flag algebra method of Razborov.

Paper Structure

This paper contains 18 sections, 15 theorems, 74 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Extremal $\lambda_{\gamma}\ifstrempty{}{}{()}$-problem
  5. Flag Algebras
  6. Individual cases
  7. Stars and matchings
  8. Exact result for $H_0$
  9. Exact result for $H_1$
  10. Graph $H_4$
  11. Graph $H_{15}$
  12. Computer-generated results for $H_9$, $H_{10}$ and $H_{11}$
  13. Graph $H_6$
  14. Graph $H_5$
  15. Symmetrizable problems
  16. ...and 3 more sections

Key Result

Theorem 3.3

Let $H$ denote $H_0$, which is the graph on $[4]$ with a red edge $02$ and a blue edge $13$. Take any $n\geqslant 4$. Define $k_0:=\lfloor x_0\rfloor$, where Then $\Lambda_{H}\ifstrempty{n}{}{(n)}=\Lambda_{H}\ifstrempty{QC(n,\binom{k_0+1}2)}{}{(QC(n,\binom{k_0+1}2))}$. Moreover, if $2n^2-10n+13$ is not a square then $QC(n,\binom{k_0+1}2)$ and its complement are the only extremal graphs; otherwise

Theorems & Definitions (23)

  • Conjecture 1.1
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Claim 3.6
  • Claim 3.7
  • Claim 3.8
  • Theorem 3.9
  • Theorem 3.10
  • Theorem 3.11
  • ...and 13 more