Semi-inducibility of 4-vertex graphs

TL;DR

This paper investigates the maximum number of specific edge-coloured 4-vertex graphs that can be embedded into larger graphs, using advanced combinatorial methods and computational tools to nearly resolve all cases.

Contribution

It provides a comprehensive analysis of 4-vertex non-complete graphs' semi-inducibility, resolving most cases with some exceptions, employing flag algebra techniques and computational proofs.

Findings

Resolved all but one case of 4-vertex non-complete graphs.

Applied flag algebra method for computational proofs.

Provided bounds and characterizations for semi-inducibility.

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.