Abstract
We show that the smallest class of labelled copies of an unlabelled tree T_{s,t} with 2<=s<=t, in the complete bipartite graph K_{s,t} has size st, is unique and has representative unlabelled tree the double-star, D_{s,t}. Equivalently, the tree T_{s,t} which has largest automorphism group size amongst all such trees is D_{s,t} with automorphism group size (s-1)!(t-1)! and automorphism group Sym(s-1) x Sym(t-1). Slight modications of these statements are needed if s = t. We also produce a novel method for finding all labelled copies of tree T_{s,t}.
Recommended Citation
McSorley, John. "Smallest labelled class and largest automorphism group of a tree T_{s,t}." Bulletin of the Institute of Combinatorics and its Applications 77 (Jan 2016): 71-84.