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

Monochromatic trees in random tournaments

We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order O(n/ \sqrt{log n}. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor

Trees with few leaves in tournaments

We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving the previously best known bound of $n+O(k^2)$ vertices to give a result tight up to the value of $C$. Furthermore, we show that, for each $k$, there exists $n_0$, such that, whenever $n\geqslant n_0$, any $(n+k-2)$-vertex tournament contains a copy of every $n$-vertex oriented tree with at most $k$ leaves, confirming a conjecture of Dross and Havet.Comment: 22 pages, 3 figure

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

Impartial digraphs

We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.Comment: 15 page