Regularity and separation for Sierpi\'nski products of graphs

TL;DR

This paper investigates the connectivity and regularity properties of Sierpiński products of graphs, providing necessary and sufficient conditions, and classifies regular polyhedral and planar Sierpiński products, including a study of special planar graphs.

Contribution

It offers the first complete classification of regular polyhedral and planar Sierpiński products, and introduces a new class of planar graphs with specific vertex-colouring properties.

Findings

Characterisation of higher connectivity conditions for Sierpiński products

Complete classification of regular polyhedral Sierpiński products

Complete classification of regular, connected, planar Sierpiński products

Abstract

The Sierpi\'nski product of graphs generalises the vast and relevant class of Sierpi\'nski-type graphs, and is also related to the classic lexicographic product of graphs. Our first main results are necessary and sufficient conditions for the higher connectivity of Sierpi\'nski products. Among other applications, we characterise the polyhedral ($3$-connected and planar) Sierpi\'nski products of polyhedra. Our other main result is the complete classification of the regular polyhedral Sierpi\'nski products, and more generally of the regular, connected, planar Sierpi\'nski products. To prove this classification, we introduce and study the intriguing class of planar graphs where each vertex may be assigned a colour in such a way that each vertex has neighbours of the same set of colours and in the same cyclic order around the vertex. We also completely classify the planar lexicographic products.