Separation numbers of trees

Abstract

Let G be a graph on n vertices. Given a bijection f : V ( G ) → { 1 , 2 , … , n } , let | f | = min { | f ( u ) − f ( v ) | : u v ∈ E ( G ) } . The separation number s ( G ) (also known as antibandwidth [T. Calamoneri, A. Massini, L. Török, I. Vrt’o, Antibandwidth of Complete k -ary trees, Electronic Notes in Discrete Mathematics 24 (2006), 259–266; A. Raspaud, H. Schroder, O. Sykora, L. Török, I. Vrt’o, Antibandwidth and cyclic antibandwidth of meshes and hypercubes, Discrete Mathematics 309 (2009) 3541–3552] of G is then max { | f | } over all such bijections f of G . We study the case when G is a forest, obtaining the following results. 1. Let F be a forest in which each component is a star. Then s ( F ) = n − μ 2 , where μ is the minimum value of ‖ X | − | Y ‖ over all bipartitions ( X , Y ) of F . 2. Let d be the maximum degree of a tree T on n vertices. Then (a) s ( T ) ≥ n 2 − c 1 n d , and (b) s ( T ) ≥ n 2 − c 2 d 2 log d n , where c 1 and c 2 are absolute constants. We give constructions showing that the bound (a) is asymptotically tight when d is in the range n 1 3 < d ≤ n 12 , while (b) is asymptotically tight when d is in the range n q ≤ d ≤ n 1 3 , where 0 < q < 1 3 is any fixed constant, and when d ≥ 4 is an absolute constant. We also show that for h ≥ 3 and odd d ≥ 3 , we have s ( T h d ) = n 2 − Θ ( d 2 + d h ) , where T h d is the symmetric d -ary tree of height h , improving the estimates obtained in the first of the above-mentioned references.