Rooted Spanning Trees in Tournaments

Abstract

A tournament of order n is an orientation of a complete graph with n vertices, and is speci®ed by its vertex set V T and edge set ET . A rooted tree is a directed tree such that every vertex except the root has in-degree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature. This result was strengthened to the following one: Every tournament contains a rooted spanning 2-tree of depth at most 2. We prove that every tournament of order n 800 contains a spanning rooted special 2-tree in this paper, where a rooted special 2-tree is a rooted 2-tree of depth 2 such that all except possibly one, non-root, non-leaf vertices, have out-degree 2 in the tree. A Short SurveyA tournament of order n is an orientation of a complete graph with n vertices, and is speci®ed by its vertex set V T and edge set ET . For an edge ab e ET , we also write a 3 b. When T is clear, we will simply write V and E for short. For A t V and B t V , we set AY B T fab e E j a e AY b e Bg.A rooted tree is a directed tree such that every vertex except the root has indegree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. A rooted 1-tree is also called a claw. The degree of a rooted tree is the out-degree of the root. For example, a claw of degree 1 is just a directed path. The depth of a rooted tree is the length of a longest directed path starting from the root.It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature.