Exact Counts of $C_{4}$s in Blow-Up Graphs

TL;DR

This paper derives exact counts of 4-cycles in blow-up graphs, a method used to maximize subgraph counts, focusing on cycles and generalized theta graphs.

Contribution

It provides the first precise enumeration of 4-cycles in blow-up graphs applied to cycles and theta graphs, advancing understanding of subgraph maximization.

Findings

Exact counts of 4-cycles in blow-up graphs are established.

Results apply to cycles and generalized theta graphs.

Enhances methods for subgraph maximization in graph theory.

Abstract

Cycles have many interesting properties and are widely studied in many disciplines. In some areas, maximising the counts of $k$-cycles are of particular interest. A natural candidate for the construction method used to maximise the number of subgraphs $H$ in a graph $G$, is the \emph{blow-up} method. Take a graph $G$ on $n$ vertices and a pattern graph $H$ on $k$ vertices, such that $n\geq k$, the blow-up method involves an iterative process of replacing vertices in $G$ with a copy of the $k$-vertex graph $H$. In this paper, we apply the blow-up method on the family of cycles. We then present the exact counts of cycles of length 4 for using this blow-up method on cycles and generalised theta graphs.