On avoidable and unavoidable trees

Abstract

A directed tree is a rooted tree if there is one vertex (the root) of in-degree 0 and every other vertex has in-degree 1. The depth of a rooted tree is the length of a longest path from the root. A directed graph G is called n-unavoidable if every tournament of order n contains it as a subgraph. M. Saks and V. Sos [“On Unavoidable Subgraphs of Tournaments,” Colloquia Mathematica Societatis Janos Bolyai 37, Finite and Infinite Sets, Eger, Hungary (1981), 663–674] constructed unavoidable rooted spanning trees of depth 3. There they wrote, “It is natural to ask how small the depth of a spanning n-unavoidable rooted tree can be.” In this paper we construct unavoidable rooted spanning trees of depth 2. Note that the depth 2 is the best we can do. For each n define λ(n) to be the largest real number such that every claw with degree d ≤ λ(n)n is n-unavoidable. The example in X. Lu [“On Claws Belonging to Every Tournament,” Combinatorica, Vol. 11 (1991), pp. 173–179] showed that λ(n) < 1/2 for sufficiently large n, but the upper bound on λ(n) given there tends to 1/2 for large n. Let λ be the lim sup of λ(n) as n tends to infinity. In this paper we show that λ is strictly less than 1/2, specifically λ ≤ 25/52. © 1996 John Wiley & Sons, Inc.