Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The semi-inducibility problem

Abdul Basit, Bertille Granet, Daniel Horsley, André Kündgen, Katherine Staden

TL;DR

The paper introduces and develops the semi-inducibility problem for red-blue edge-coloured complete graphs, focusing on maximizing copies of a fixed red-blue graph H in K_n. It delivers sharp or near-sharp bounds for alternating walks, alternating cycles (including 4t-cycles) and several non-alternating 4-cycle patterns, showing extremals are often bipartite or quasirandom depending on density; it also constructs a quantum graph with positive coefficients that yields an interval of densities where the binomial random graph is asymptotically extremal and unique up to quasirandomness. The authors connect these results to the inducibility problem, Ramsey multiplicity, and the feasible region of quantum graphs, and develop stability tools and partitioned/canonical graph concepts to characterise extremal configurations. The work highlights when randomness can be an extremal guide and when structured partitions dominate, providing a framework and open questions for further exploration in semi-inducibility and quantum-graph inducibility. Overall, the results deepen our understanding of how colour patterns constrain subgraph counts and reveal intricate links between extremal structure and probabilistic randomness in coloured settings.

Abstract

Let $H$ be a $k$-edge-coloured graph and let $n$ be a positive integer. What is the maximum number of copies of $H$ in a $k$-edge-coloured complete graph on $n$ vertices? This paper studies the case $k=2$, which we call the semi-inducibility problem. This problem is a generalisation of the inducibility problem of Pippenger and Golumbic which is solved only for some small graphs and limited families of graphs. We prove sharp or almost sharp results for alternating walks, for alternating cycles of length divisible by 4, and for 4-cycles of every colour pattern. Liu, Mubayi and Reiher asked whether there is a graph $F$ for which the binomial random graph is an asymptotically extremal graph in the inducibility problem over all graphs of a given edge density. This was recently answered in a strong negative sense by Jain, Michelen and Wei. In contrast, we find a \emph{quantum} graph $Q$ with positive coefficients and an interval of edge densities for which the only extremal graphs are quasirandom.

The semi-inducibility problem

TL;DR

The paper introduces and develops the semi-inducibility problem for red-blue edge-coloured complete graphs, focusing on maximizing copies of a fixed red-blue graph H in K_n. It delivers sharp or near-sharp bounds for alternating walks, alternating cycles (including 4t-cycles) and several non-alternating 4-cycle patterns, showing extremals are often bipartite or quasirandom depending on density; it also constructs a quantum graph with positive coefficients that yields an interval of densities where the binomial random graph is asymptotically extremal and unique up to quasirandomness. The authors connect these results to the inducibility problem, Ramsey multiplicity, and the feasible region of quantum graphs, and develop stability tools and partitioned/canonical graph concepts to characterise extremal configurations. The work highlights when randomness can be an extremal guide and when structured partitions dominate, providing a framework and open questions for further exploration in semi-inducibility and quantum-graph inducibility. Overall, the results deepen our understanding of how colour patterns constrain subgraph counts and reveal intricate links between extremal structure and probabilistic randomness in coloured settings.

Abstract

Let be a -edge-coloured graph and let be a positive integer. What is the maximum number of copies of in a -edge-coloured complete graph on vertices? This paper studies the case , which we call the semi-inducibility problem. This problem is a generalisation of the inducibility problem of Pippenger and Golumbic which is solved only for some small graphs and limited families of graphs. We prove sharp or almost sharp results for alternating walks, for alternating cycles of length divisible by 4, and for 4-cycles of every colour pattern. Liu, Mubayi and Reiher asked whether there is a graph for which the binomial random graph is an asymptotically extremal graph in the inducibility problem over all graphs of a given edge density. This was recently answered in a strong negative sense by Jain, Michelen and Wei. In contrast, we find a \emph{quantum} graph with positive coefficients and an interval of edge densities for which the only extremal graphs are quasirandom.
Paper Structure (28 sections, 42 theorems, 162 equations, 1 table)

This paper contains 28 sections, 42 theorems, 162 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Ramsey multiplicity and the inducibility problem
  4. Goodman's bound and Ramsey theory
  5. The inducibility problem
  6. Random graphs as maximisers
  7. Further related work
  8. Known results on the semi-inducibility problem
  9. Preliminaries
  10. Organisation and proof ideas
  11. Notation
  12. Inequalities etc.
  13. General results
  14. delta-bipartite and partitioned graphs
  15. Alternating 4-cycles
  16. ...and 13 more sections

Key Result

Theorem 1.3

Every red-blue $K_n$ has at most $2n\left(\frac{n-1}{2}\right)^{t}$ alternating walks of length $t \ge 1$. Equality is attained for colourings in which every vertex is incident with $\frac{n-1}{2}$ red and $\frac{n-1}{2}$ blue edges.

Theorems & Definitions (76)

  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8: Quasirandom
  • Theorem 1.9
  • Theorem 2.1: Goodman Goodman59
  • Theorem 2.4: Schelp and Thomason SchelpThomason98
  • Theorem 2.5: Jain2023binomial
  • ...and 66 more