Efficient Enumeration of Cliques in Graphs with Bounded Maximum Degree

TL;DR

This paper introduces a new method for counting cliques in graphs with bounded maximum degree, providing a novel proof of existing theorems and advancing understanding of extremal graph enumeration problems.

Contribution

It presents an innovative approach for counting cliques and offers a new proof of key extremal theorems related to graphs with bounded degree.

Findings

New proof of the Cutler-Radcliffe theorem

New proof of the Kahn-Zhao theorem

Enhanced understanding of clique enumeration in bounded degree graphs

Abstract

In recent years, there has been a surge of interest in extremal problems concerning the enumeration of independent sets or cliques in graphs with specific constraints. For instance, the Kahn-Zhao theorem establishes an upper bound on the number of independent sets in a $d$-regular graph. Building on this, Cutler and Radcliffe extended the result by identifying the graph that maximizes the number of cliques among graphs with bounded order and maximum degree. In this paper, we introduce an innovative approach for counting cliques in graphs with a bounded maximum degree. To demonstrate the effectiveness of the method, we provide a new proof for the above Cutler-Radcliffe theorem and the Kahn-Zhao theorem.