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On the cop number and the weak Meyniel conjecture for algebraic graphs

Arindam Biswas, Jyoti Prakash Saha

TL;DR

This work analyzes the cops-and-robbers pursuit on algebraic graphs derived from finite groups, focusing on Cayley sum graphs, generalized Cayley graphs, twisted Cayley graphs, and twisted Cayley sum graphs. It establishes a uniform bound of $c\le 2|S|$ for undirected, connected instances (under the stated symmetry/conjugation conditions) by a coordinated strategy using primary and auxiliary cops and a tail/power reduction mechanism. By applying Bollobás–Janson–Riordan’s results, the authors further deduce a Meyniel-type bound $c=O(n^{1-\varepsilon})$ (indeed $c\le 2\sqrt{n}\log n$) for these algebraic graphs, thereby validating the weak Meyniel conjecture in this broad class. The findings extend Frankl’s bound for Cayley graphs and illuminate pursuit-evasion dynamics on structured, highly symmetric graphs with applications to algebraic graph theory.

Abstract

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop number of generalised Cayley graphs and twisted Cayley sum graphs under some conditions. These extend a result of Frankl to such graphs. Using the above bounds and a result of Bollobás--Janson--Riordan, we show that the weak Meyniel conjecture holds for these algebraic graphs.

On the cop number and the weak Meyniel conjecture for algebraic graphs

TL;DR

This work analyzes the cops-and-robbers pursuit on algebraic graphs derived from finite groups, focusing on Cayley sum graphs, generalized Cayley graphs, twisted Cayley graphs, and twisted Cayley sum graphs. It establishes a uniform bound of for undirected, connected instances (under the stated symmetry/conjugation conditions) by a coordinated strategy using primary and auxiliary cops and a tail/power reduction mechanism. By applying Bollobás–Janson–Riordan’s results, the authors further deduce a Meyniel-type bound (indeed ) for these algebraic graphs, thereby validating the weak Meyniel conjecture in this broad class. The findings extend Frankl’s bound for Cayley graphs and illuminate pursuit-evasion dynamics on structured, highly symmetric graphs with applications to algebraic graph theory.

Abstract

We show that the cop number of the Cayley sum graph of a finite group with respect to a symmetric subset is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop number of generalised Cayley graphs and twisted Cayley sum graphs under some conditions. These extend a result of Frankl to such graphs. Using the above bounds and a result of Bollobás--Janson--Riordan, we show that the weak Meyniel conjecture holds for these algebraic graphs.
Paper Structure (25 sections, 6 theorems, 36 equations)

This paper contains 25 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Algebraic graphs
  3. Results obtained
  4. Strategy for bounding the cop number
  5. The moves of the primary cops
  6. Notations
  7. The strategy
  8. Reduction of tail and power
  9. Cop numbers of Cayley sum graphs
  10. The moves of the primary cops
  11. Reduction of tail and power
  12. The moves of the auxiliary cops
  13. Capturing the robber
  14. Cop numbers of twisted Cayley graphs
  15. The moves of the primary cops
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose the Cayley sum graph $C_\Sigma(G, S)$ is undirected and connected. Assume that $S$ is symmetric. Then the cop number of $C_\Sigma(G, S)$ is at most $2|S|$.

Theorems & Definitions (13)

  • Theorem 1.1: Cop number of undirected Cayley sum graphs
  • Theorem 1.2: Cop number of undirected twisted Cayley graphs
  • Theorem 1.3: Cop number of undirected twisted Cayley sum graphs
  • Theorem 1.4: \ref{['Thm:WMC']}
  • Lemma 2.1
  • proof : Proof of \ref{['Lemma:ReductionGen']} for $\Gamma = C_\Sigma(G, S)$
  • proof : Proof of \ref{['Thm:CayleySumCopNoBdd']}
  • proof : Proof of \ref{['Lemma:ReductionGen']} for $\Gamma = C(G, S)^\sigma$
  • proof : Proof of \ref{['Thm:CayleyTwistedCopNoBdd']}
  • proof : Proof of \ref{['Lemma:ReductionGen']} for $\Gamma = C_\Sigma(G, S)^\sigma$
  • ...and 3 more