2009/10/24 by János Barát, Barát, János, Peter I. Hajnal +4
Computer Science · Mathematics · #05C05 #05D99 #Advanced Graph Theory Research #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C05 #msc:05D99 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0910.4647
11 pages, 5 figures, submitted on March 31
arxiv created 2009/10/24 · openalex publication_date 2009/10/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or disjoint. Recently, numerous extremal questions have been answered using different methods. We show elementary techniques unifying these approaches. Our building parts are based on rooted binary trees and discrete geometry. Among other things, we show the maximum cardinality of islands on a toroidal board and in a hypercube. We also strengthen a previous result by rarefying the neighborhood relation.