On Meyniel extremal families of graphs

TL;DR

This paper introduces new constructions of Meyniel extremal graphs, explores their properties, and establishes bounds on cop numbers, contributing to the understanding of graph families with extremal pursuit-evasion characteristics.

Contribution

It provides novel methods to construct Meyniel extremal graphs and analyzes their degrees, chromatic number, and diameter, expanding the known classes of such graphs.

Findings

Exponential number of new Meyniel extremal families with specified degrees.

Best-known upper bound on cop number of vertex-transitive graphs.

Identification of Meyniel extremal graphs with large chromatic number and diameter.

Abstract

We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number. Using spanning subgraphs, we prove that there are an exponential number of new Meyniel extremal families with specified degrees. Using a linear programming problem on hypergraphs, we explore the degrees in families that are not Meyniel extremal. We give the best-known upper bound on the cop number of vertex-transitive graphs with a prescribed degree. We find new Meyniel extremal families of regular graphs with large chromatic number, large diameter, and explore the connection between Meyniel extremal graphs and bipartite graphs.