Star-factors of tournaments

Let Sm denote the m-vertex simple digraph formed by m− 1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of Sm. We prove that m lgm−m lg lgm ≤ f(m) ≤ 4m − 6m for sufficiently large m. If D is an acyclic digraph of order n, then D occurs as a subgraph of every tournament with at least 2 vertices; this follows easily by induction on n. Special digraphs can be guaranteed to appear even when there are no extra vertices. A digraph of order n is unavoidable if it appears in every n-vertex tournament. A claw is a digraph obtained by identifying the sources of a set of edge-disjoint paths. The question of which n-vertex claws are unavoidable is studied in [5,6,10]. Further references and additional unavoidable directed trees appear in [7]. A claw formed using paths of length one is a star; the m-star Sm consists of m − 1 edges with a common tail and m − 1 distinct heads. We use kSm to denote a disjoint union of k stars of order m. A 3-vertex tournament need not contain a spanning S3, but every 6-vertex tournament contains a spanning 2S3. Some 8-vertex tournaments avoid 2S4 (see Theorem 3). A copy of kSm in a tournament of order km is an m-star-factor of the tournament. If m-star-factors are unavoidable for some multiple of m, then inductively they are unavoidable for all larger multiples of m. Let f(m) be the least n such that every nvertex tournament contains an m-star-factor (if such n exists). We prove that m lgm − m lg lgm ≤ f(m) ≤ 4m − 6m for sufficiently large m. The upper bound holds for all m. (Always lg denotes log2.) Reid [9] asked for the minimum n such that all n-vertex tournaments have k pairwisedisjoint transitive subtournaments of order m. For fixed m, Erdős proved the existence * Research supported in part by NSA/MSP Grant MDA904-94-H-2060. † Research supported in part by NSA/MSP Grant MDA904-93-H-3040. Running head: STAR-FACTORS OF TOURNAMENTS AMS codes: 05C20, 05C35, 05C70.