On Unavoidability of Trees with k Leaves

Abstract.

 A digraph is said to be n-unavoidable if every tournament of order n contains it as a subgraph. Let f(n) be the smallest integer such that every oriented tree of order n is f(n)-unavoidable. Sumner (see [6]) noted that f(n)≥2n−2 and conjectured that equality holds. Havet and Thomassé [4] established the upper bound . Sumner's conjecture is implied by a stronger one due to Havet and Thomassé: g(k)≤k−1 where g(k) is the smallest integer such that every oriented tree of order n with k leaves is (n+g(k)) -unavoidable. Häggkvist and Thomason [1] proved that g(k)≤2^512k3 and Havet [2] proved g(3)≤5. A node is a vertex with sum of out- and indegree at least three in an oriented tree. An oriented tree is constructible if every path from a node to another node is not directed with first and last block (maximal directed subpath) of length at least two and every path from node to a leaf has first block of length at least two. In this paper, we prove that Havet and Thomassé's conjecture is true for constructible trees.