Star-factors of tournaments

Let S_m 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 S_m. We prove that m lg m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.Comment: 5 pages, 1 figur

Trees in tournaments

AbstractA 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 is f(n)-unavoidable. Sumner (see (Reid and Wormald, Studia Sci. Math. Hungaria 18 (1983) 377)) noted that f(n)⩾2n−2 and conjectured that equality holds. Häggkvist and Thomason established the upper bounds f(n)⩽12n and f(n)⩽(4+o(1))n. Let g(k) be the smallest integer such that every oriented tree of order n with k leaves is (n+g(k))-unavoidable. Häggkvist and Thomason (Combinatorica 11 (1991) 123) proved that g(k)⩽2512k3. Havet and Thomassé conjectured that g(k)⩽k−1. We study here the special case where the tree is a merging of paths (the union of disjoint paths emerging from a common origin). We prove that a merging of order n of k paths is (n+32(k2−3k)+5)-unavoidable. In particular, a tree with three leaves is (n+5)-unavoidable, i.e. g(3)⩽5. By studying trees with few leaves, we then prove that f(n)⩽385n−6

Trees in tournaments

AbstractA 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 is f(n)-unavoidable. Sumner (see (Reid and Wormald, Studia Sci. Math. Hungaria 18 (1983) 377)) noted that f(n)⩾2n−2 and conjectured that equality holds. Häggkvist and Thomason established the upper bounds f(n)⩽12n and f(n)⩽(4+o(1))n. Let g(k) be the smallest integer such that every oriented tree of order n with k leaves is (n+g(k))-unavoidable. Häggkvist and Thomason (Combinatorica 11 (1991) 123) proved that g(k)⩽2512k3. Havet and Thomassé conjectured that g(k)⩽k−1. We study here the special case where the tree is a merging of paths (the union of disjoint paths emerging from a common origin). We prove that a merging of order n of k paths is (n+32(k2−3k)+5)-unavoidable. In particular, a tree with three leaves is (n+5)-unavoidable, i.e. g(3)⩽5. By studying trees with few leaves, we then prove that f(n)⩽385n−6

k-Ary spanning trees contained in tournaments

A rooted tree is called a $k$-ary tree, if all non-leaf vertices have exactly $k$ children, except possibly one non-leaf vertex has at most $k-1$ children. Denote by $h(k)$ the minimum integer such that every tournament of order at least $h(k)$ contains a $k$-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that $h(1)=1$. Lu et al. [J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of $h(k)$, and showed that $h(2)=4$ and $h(3)=8$. The exact values of $h(k)$ remain unknown for $k\geq 4$. A result of Erd\H{o}s on the domination number of tournaments implies $h(k)=\Omega(k\log k)$. In this paper, we prove that $h(4)=10$ and $h(5)\geq13$.Comment: 11 pages, to appear in Discrete Applied Mathematic

Oriented trees and paths in digraphs

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area

Largest Digraphs Contained IN All N-tournaments

Let f(n) (resp. g(n)) be the largest m such that there is a digraph (resp. a spanning weakly connected digraph) on n-vertices and m edges which is a subgraph of every tournament on n-vertices. We prove that n log2 n--cxn&gt;=f(n) ~_g(n) ~- n log ~ n--c..n loglog n

On Seymour's and Sullivan's Second Neighbourhood Conjectures

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.Comment: 14 pages, 1 figure