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Efficient Enumeration of Cliques in Graphs with Bounded Maximum Degree

Shi-Cai Gong, Jia-Jin Wang, Xin-Hao Zhu, Bo-Jun Yuan

TL;DR

This paper studies counting cliques in graphs with bounded maximum degree and builds on the Kahn-Zhao bound for independent sets and the Cutler-Radcliffe clique bound. It introduces a local counting framework via the vertex clique weight $w_G(u)$ and a global decomposition $k(G)=1+\sum_{u\in V(G)} w_G(u)$, complemented by edge-based counts $k(e;G)$ and inclusion–exclusion, together with Kruskal-Katona and Colex graphs to obtain tight bounds. The authors provide a new proof of both the Kahn-Zhao and Cutler-Radcliffe theorems and prove that, for graphs on $n$ vertices with maximum degree at most $r$, the maximum number of cliques is achieved by a disjoint union of $\left\lfloor \frac{n}{r+1}\right\rfloor$ copies of $K_{r+1}$. The work also characterizes small-$r$ extremals (e.g., $r=1$ and $r=2$) and offers a versatile counting framework with potential applications to subgraph counts under degree constraints.

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.

Efficient Enumeration of Cliques in Graphs with Bounded Maximum Degree

TL;DR

This paper studies counting cliques in graphs with bounded maximum degree and builds on the Kahn-Zhao bound for independent sets and the Cutler-Radcliffe clique bound. It introduces a local counting framework via the vertex clique weight and a global decomposition , complemented by edge-based counts and inclusion–exclusion, together with Kruskal-Katona and Colex graphs to obtain tight bounds. The authors provide a new proof of both the Kahn-Zhao and Cutler-Radcliffe theorems and prove that, for graphs on vertices with maximum degree at most , the maximum number of cliques is achieved by a disjoint union of copies of . The work also characterizes small- extremals (e.g., and ) and offers a versatile counting framework with potential applications to subgraph counts under degree constraints.

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 -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.
Paper Structure (3 sections, 11 theorems, 48 equations)

This paper contains 3 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['maina']}

Key Result

Theorem 1.1

(Kahn-Zhao) If $G$ is a $d$-regular graph with $n$ vertices, then where $i(G)$ denotes the number of independent sets contained in the graph $G$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 14 more