On disjoint configurations in infinite graphs

Abstract

AbstractFor a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ℵ0 disjoint copies of A is referred to as ℵ0A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all nϵℕ, but ℵ0A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G1, G2,… such that Gi is not a minor of Gj for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ϵ ℕ, then ℵ0A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016