Infinite benzenoids

Abstract

A family of benzenoids, called convex benzenoids, was introduced in 2012 by Cruz, Gutman and Rada. In a later paper by the present author et al., several equivalent characterisations of convex benzenoids have been given and their equivalence was proved. Along the way an infinite benzenoid called the half-plane was used for the purpose of theoretical reasoning. In this short paper, some properties of infinite benzenoids are discussed. It is proved that their boundary consists of countably many connected components. Convex infinite benzenoids are classified and it is proved that there are only countably many convex infinite benzenoids, whilst there are uncountably many infinite (non-convex) benzenoids. We also show that there are countably many infinite benzenoids which have a finite number of 1s in their boundary-edges code.