Unfriendly partitions when avoiding vertices of finite degree
Leandro Fiorini Aurichi,
Lucas Silva Sinzato Real
Abstract:An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by Cowan and Emerson [2] that every graph has an unfriendly partition, but Milner and Shelah in [5] found counterexamples for that statement by analysing graphs with uncountably many vertices. Curiously, none of their graphs have vertices with finite degree. Therefore, as a natural… Show more
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.