相关论文: Paths of given length in tournaments

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is…

A vertex subset of a graph is called a distance-$k$ independent set if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal…

The Traveling Tournament Problem is a sports-scheduling problem where the goal is to minimize the total travel distance of teams playing a double round-robin tournament. The constraint 'k' is an imposed upper bound on the number of…

We study the following Ramsey-theoretic question: given a $q$-coloring of the edges of a tournament, how long of a directed path can we guarantee whose edges avoid one of the colors? Questions of this type have applications in many areas,…

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an…

A digraph $D$ is $k$-linked if for every $2k$ distinct vertices $ x_1,\ldots , x_k, y_1, \ldots , y_k$ in $D$, there exist $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ starts at $x_i$ and ends at $y_i$ for each $i\in…

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common…

The Traveling Tournament Problem (TTP-$k$) is a well-known benchmark problem in tournament timetabling, which asks us to design a double round-robin schedule such that the total traveling distance of all $n$ teams is minimized under the…

We form a "map of tournaments" by adapting the map framework from the world of elections. By a tournament we mean a complete directed graph where the nodes are the players and an edge points from a winner of a game to the loser (with no…

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for…

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…

Havet and Thomass\'{e} proved that every tournament of order $n\geq 8$ contains every oriented Hamiltonian path, which was conjectured by Rosenfeld. Recently, it was shown that in any tournament $T$ of order $n\geq 8$, there exists an arc…

A king in a directed graph is a vertex $v$ such that every other vertex is reachable from $v$ via a path of length at most $2$. It is well known that every tournament (a complete graph where each edge has a direction) has at least one king.…

Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This…

Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Guti\'errez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon…

Based on known methods for computing the number of distinct score sequences for $n$-vertex tournaments, we develop algorithms for computing the number of distinct score sequences for self-complementary tournaments, strong tournaments, and…

Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree…

If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$…

We prove that there exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integer $k$, if $T$ is a strongly $4k$-connected tournament with minimum out-degree at least $f(k)$, then $T$ is $k$-linked. This makes…

In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at…