Maximizing the number of independent sets of fixed size in $K_n$-covered graphs

TL;DR

This paper determines the maximum number of independent sets of a fixed size in graphs covered by complete graphs, providing an extremal graph construction and answering a previously posed question.

Contribution

It introduces a method to find the maximum independent sets in $K_n$-covered graphs and identifies the extremal graph structure for this problem.

Findings

Maximum number of independent sets of size t in $K_n$-covered graphs is established.

Extremal graph characterized for maximizing independent sets.

Edge-switching operation preserves the count of independent sets.

Abstract

A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine its extremal graph. The result answers a question proposed by Chakraborit and Loh. The proof uses an edge-switching operation of hypergraphs which remains the number of independent sets nondecreasing.