The Tree Alternative Conjecture concerns the sizes of equivalence classes of trees with respect to mutual embeddable relation. Indeed, it conjectures that the number of isomorphism classes of trees mutually embeddable with a given tree $T$ is either 1 or infinite - with instances of size $\aleph_0$ and $2^{\aleph_0}$. We prove its analogue within the family of locally finite trees with respect to the topological minor relation. More precisely, we prove that for any locally finite tree $T$ the size of its equivalence class with respect to the topological minor relation can only be either $1$ or $2^{\aleph_0}$.
@article{10_37236_10556,
author = {Jorge Bruno and Paul Szeptycki},
title = {The tree alternative conjecture under the topological minor relation},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10556},
zbl = {1486.05202},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10556/}
}
TY - JOUR
AU - Jorge Bruno
AU - Paul Szeptycki
TI - The tree alternative conjecture under the topological minor relation
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10556/
DO - 10.37236/10556
ID - 10_37236_10556
ER -
%0 Journal Article
%A Jorge Bruno
%A Paul Szeptycki
%T The tree alternative conjecture under the topological minor relation
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10556/
%R 10.37236/10556
%F 10_37236_10556
Jorge Bruno; Paul Szeptycki. The tree alternative conjecture under the topological minor relation. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10556
