A non-aligning variant of generalized Tur\'an problems

TL;DR

This paper introduces a new variant of generalized Turán problems where the restriction is on the placement of copies of graphs rather than their absence, and provides solutions and bounds for specific cases.

Contribution

It defines and analyzes a non-aligning variant of generalized Turán problems, offering solutions and bounds for certain graph pairs and applying results to determine specific Turán numbers.

Findings

Solved the problem for some graph instances.

Provided bounds for other cases.

Determined generalized Turán numbers for certain graph pairs.

Abstract

In the so-called generalized Tur\'an problems we study the largest number of copies of $H$ in an $n$-vertex $F$-free graph $G$. Here we introduce a variant, where $F$ is not forbidden, but we restrict how copies of $H$ and $F$ can be placed in $G$. More precisely, given an integer $n$ and graphs $H$ and $F$, what is the largest number of copies of $H$ in an $n$-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of $F$? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the generalized Tur\'an number for some pairs of graphs.