Approximating Cayley diagrams versus Cayley graphs

TL;DR

The paper explores the differences between Cayley diagrams and Cayley graphs through graph sequences, highlighting limitations in edge labelling convergence and implications for Hamiltonian cycles and property testing.

Contribution

It constructs graph sequences illustrating the divergence between Cayley diagrams and graphs, and examines the testability of Hamiltonian cycles in this context.

Findings

Sequences of graphs can converge to Cayley graphs without their Cayley diagrams converging.

Certain spanning trees, like Hamiltonian cycles, may not be approximable by subgraphs in converging sequences.

The results relate to the testability of Hamiltonian cycles as a graph property.

Abstract

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a spanning tree in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this subtree is a Hamiltonian cycle, but convergence is meant in a stronger sense. These latter are related to whether having a Hamiltonian cycle is a testable graph property.