An oriented tree $T$ on $n$ vertices is unavoidable if every tournament on $n$ vertices contains a copy of $T$. In this paper we give a sufficient condition for $T$ to be unavoidable, and use this to prove that almost all labelled oriented trees are unavoidable, verifying a conjecture of Bender and Wormald. We additionally prove that every tournament on $n + o(n)$ vertices contains a copy of every oriented tree $T$ on $n$ vertices with polylogarithmic maximum degree, improving a result of Kühn, Mycroft and Osthus.
For a set H of tournaments, we say H is heroic if every tournament, not containing any member of H as a subtournament, has bounded chromatic number. Berger et al. explicitly characterized all heroic sets containing one tournament. Motivated by this result, we study heroic sets containing two tournaments. We give a necessary condition for a set containing two tournaments to be heroic. We also construct infinitely many minimal heroic sets of size two.
We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for $t \ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $δ$-far from being monochromatic contains an \emph{unavoidable $t$-colouring} when $δ\gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a clique of order $2t$ in which one colour forms either a clique of order $t$ or two disjoint cliques of order $t$. Next, for $ t\ge 3$, we show that any tournament on $n$ vertices that is $δ$-far from being transitive contains an \emph{u…
Let $D_k$ denote the tournament on $3k$ vertices consisting of three disjoint vertex classes $V_1, V_2$ and $V_3$ of size $k$, each of which is oriented as a transitive subtournament, and with edges directed from $V_1$ to $V_2$, from $V_2$ to $V_3$ and from $V_3$ to $V_1$. Fox and Sudakov proved that given a natural number $k$ and $ε> 0$ there is $n_0(k,ε)$ such that every tournament of order $n_0(k,ε)$ which is $ε$-far from being transitive contains $D_k$ as a subtournament. Their proof showed that $n_0(k,ε) \leq ε^{-O(k/ε^2)}$ and they conjectured that this could be reduced to $n_0(k,ε) \leq…
Given a tournament T, let h(T) be the smallest integer k such that every arc-coloring of T with k or more colors produces at least one out-directed spanning tree of T with no pair of arcs with the same color. In this paper we give the exact value of h(T).
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.
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 generalises a result of the first, third and forth authors who proved the same statement for paths, and is tight up to a constant factor.
Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains a copy of any directed tree on $n$ vertices. We prove an asymptotic version of this conjecture, namely that any tournament on $(2+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed $Δ$, any tournament on $(1+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices with maximum degree at most $Δ$.
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 Erdős on the domination number o…
A shortcut of a directed path $v_1 v_2 \cdots v_n$ is an edge $v_iv_j$ with $j > i+1$. If $j = i+2$ the shortcut is called a hop. If all hops are present, the path is called hop complete, so the path and its hops form a square of a path. We prove that every tournament with $n \ge 4$ vertices has a Hamiltonian path with at least $(4n-10)/7$ hops, and has a hop complete path of order at least $n^{0.295}$. A spanning binary tree of a tournament is a spanning shortcut tree if for every vertex of the tree, all its left descendants are in-neighbors and all its right descendants are out-neighbors. It…
Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds for all sufficiently large $n$. The proof makes extensive use of results and ideas from a recent paper by the same authors, in which an approximate version of the conjecture was proved.
A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\geq 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k -2)$-unavoidable and $(\frac{9}{2}n -\frac{5}{2}k -\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8} n- \frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+ 144k^2 - 280k + 124)$-unavoidable.
The classical paradox of social choice theory asserts that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on N vertices (one per candidate), placing an edge from U to V if U would beat V in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a complete binary tree whose leaves are labele…
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least |T|n and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-v…
In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $ \overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Rams…
Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).
A graph has tree-width at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums. We refine this definition by, for each non-negative integer $θ$, defining the $θ$-tree-width of a graph to be at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums on cliques of size less than $θ$. We find the unavoidable minors for the graphs with large $θ$-tree-width and we obtain Robertson and Seymour's Grid Theorem as a corollary.
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is $k$-colourable if and only if it admits an orientation with no directed path on more than $k$-vertices. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a dual…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we…
