Simplicial Decompositions of Infinite Graphs

Abstract

Simplicia1 decompositions are used for the characterizations of all (finite) maximal graphs not contractible on K 5 and, respectively, on K 3,3 . In the case of infinite graphs, simplicia1 decompositions are a useful tool. Under general assumptions, an uncountable graph G has simplicial decomposition whose members are all of “small” cardinality. This made it, for instance, possible to tackle a generalization of Hadwiger's conjecture to graphs with infinite chromatic number. This chapter presents a general decomposition theorem that is applied to several problems. For example, to the structure of the graphs, which do not contain a subdivision of a complete graph of a given uncountable order. The chapter discusses separation-invariant subgraphs. Every (induced) subgraph H of G that occurs as a member in some simplicial decomposition of G is called a simplicial summand of G .