We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal construction. It is an unbalanced blow-up of an edge, where the sink vertex is replaced by a quasi-random tournament and the source vertex is iteratively replaced by a copy of the construction itself.
In 1975, Pippenger and Golumbic conjectured that every n-vertex graph has at most $n^k/(k^k - k)$ induced cycles of length k for k at least 5. We prove that every n-vertex graph has at most $2 n^k/k^k$ induced cycles of length k.
In the area of extremal graph theory, there exists a problem that investigates the maximum induced density of a $k$-vertex graph $H$ in any $n$-vertex graph $G$. This is known as the problem of \emph{inducibility} that was first introduced by Pippenger and Golumbic in 1975. In this paper, we give a new upper bound for the inducibility for a family of \emph{Double Loop Graphs} of order $k$. The upper bound obtained for order $k=5$ is within a factor of 0.964506 of the exact inducibility, and the upper bound obtained for $k=6$ is within a factor of 3 of the best known lower bound.
Given a $k$-vertex graph $H$ and an integer $n$, what are the $n$-vertex graphs with the maximum number of induced copies of $H$? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of $k$-vertex subsets of an $n$-vertex graph that induce a copy of $H$. Huang, Lee and the first author proved that for a random $k$-vertex graph $H$, almost surely the $n$-vertex graphs maximizing the number of induced copies of $H$ are the balanced iterated blow-ups of $H$. In this paper, we consider the case where…
Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For $T$ in the family of strictly $d$-ary trees (every vertex has $0$ or $d$ children), we prove that the difference between the maximum density of a $d$-ary tree $D$ in $T$ and the inducibility of $D$ is of order $\mathcal{O}(|T|^{-1/2})$ compared to the general case where it is shown that the difference is $\…
A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. The conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. Both the directed and the undirected versions of the conjecture are known to be…
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result t…
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.
A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament…
Sullivan stated the conjectures: (1) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)\geq d^{-}(x)$; (2) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)+d^{+}(x)\geq 2d^{-}(x)$. In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament $D$, we prove that $D$ has at least two vertices satisfying $(1)$ if $D$ has no vertex of in-degree zero. And, for a local tournament $D$, we prove that either there exist two vertices satisfying $(2)$ or there exists a vertex $v$ satisfying $d^{++}(v)+d^{+}(v)\geq 2d^{-}(v)+2$ i…
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.
It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge.
The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term. We also show that if $T$ is an $n$-vertex tournament with $Δ^+(T)-δ^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. 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 c|T| and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tourn…
For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this is the expected density of $H$ in a random tournament. A family ${\mathcal F}$ of $h$-vertex tournaments is {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex tournament $G$ such that the density of each element of ${\mathcal F}$ in $G$ is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small $h$. Here we characterize several large dominant families for every $h$. In particular, we prove the follow…
In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a {\em tournament} is an orientation of a complete graph. In addition, we show that every arc-colored $n$-vertex tournament with all its strongly connected $k$-vertex subtournaments, $3\leq k\leq n$, colored with at least…
Let $H$ and $K$ be subsets of the vertex set $V(Q_d)$ of the $d$-cube $Q_d$ (we call $H$ and $K$ configurations in $Q_d$). We say $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ to $K$. If $d$ is a positive integer and $H$ is a configuration in $Q_d$, we define $π(H,d)$ to be the limit as $n$ goes to infinity of the maximum fraction, over all subsets $S$ of $V(Q_n)$, of sub-$d$-cubes of $Q_n$ whose intersection with $S$ is an exact copy of $H$. We determine $π(C_8,4)$ and $π(P_4,3)$ where $C_8$ is a "perfect" 8-cycle in $Q_4$ and $P_4$ is a "perfect" pat…
Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash and Sudakov proved that if $G$ is an orientation of a graph on $n$ vertices in which every vertex has both indegree and outdegree at least $(1/2 - o(1))n$, then there exists a collection of vertex-disjoint cyclic triangles that covers all but at most $3$ vertices. In this paper, we resolve the conjecture of Cuckler and Yuster for sufficiently large $n$.
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…
Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $\vec{T}_k$'s.
Let $T_{n}$ be an arc-colored tournament of order $n$. The maximum monochromatic indegree $Δ^{-mon}(T_{n})$ (resp. outdegree $Δ^{+mon}(T_{n})$) of $T_{n}$ is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of $T_{n}$. The irregularity $i(T_{n})$ of $T_{n}$ is the maximum difference between the indegree and outdegree of a vertex of $T_{n}$. A subdigraph $H$ of an arc-colored digraph $D$ is called rainbow if each pair of arcs in $H$ have distinct colors. In this paper, we show that each vertex $v$ in an arc-colored tournament $T_{n}$ with $Δ^{-mon}(T_n)\leqΔ^{…
We show that performing just one round of the Sherali-Adams hierarchy gives an easy 7/3-approximation algorithm for the Feedback Vertex Set (FVST) problem in tournaments. This matches the best deterministic approximation algorithm for FVST due to Mnich, Williams, and Végh, and is a significant simplification and runtime improvement of their approach.
A digraph is {\bf eulerian} if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. A digraph is {\bf semicomplete} if it has no pair of non-adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. Fraise and Thomassen \cite{fraisseGC3} proved that every $(k+1)$-strong tournament has a hamiltonian cycle which avoids any prescribed set of $k$ arcs. In \cite{bangsupereuler} the authors demonstrated that a number of results concerning vertex-c…
