Related papers: Classifying Intrinsically Linked Tournaments by Sc…
A tournament is an orientation of a graph. Vertices are players and edges are games, directed away from the winner. Kannan, Tetali and Vempala and McShine showed that tournaments with given score sequence can be rapidly sampled, via simple…
We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim,…
In a directed graph, the imbalance of a vertex is its outdegree minus its indegree. We characterize the sequences that are realizable as the sequence of imbalances of a simple directed graph. Moreover, a realization of a realizable sequence…
For an orientation $H$ with $n$ vertices, let $T(H)$ denote the maximum possible number of labeled copies of $H$ in an $n$-vertex tournament. It is easily seen that $T(H) \ge n!/2^{e(H)}$ as the latter is the expected number of such copies…
This note describes necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a connected simple graph. Conditions are also given under which a sequence is necessarily connected i.e. the sequence…
We study the complexity of counting and finding small tournament patterns inside large tournaments. Given a fixed tournament $T$ of order $k$, we write ${\#}\text{IndSub}_{\text{To}}(\{T\})$ for the problem whose input is a tournament $G$…
An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or…
We consider a random knockout tournament among players $1, \ldots, n$, in which each match involves two players. The match format is specified by the number of matches played in each round, where the constitution of the matches in a round…
For an integer $k \ge 2$, let $A$ be a Boolean block matrix with blocks $A_{ij}$ for $1 \le i,j \le k$ such that $A_{ii}$ is a zero matrix and $A_{ij}+A_{ji}^T$ is a matrix with all elements $1$ but not both corresponding elements of…
We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $\Delta(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic…
Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…
We find the minimal number of links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for…
A {\em bipartite tournament} is a directed graph $T:=(A \cup B, E)$ such that every pair of vertices $(a,b), a\in A,b\in B$ are connected by an arc, and no arc connects two vertices of $A$ or two vertices of $B$. A {\em feedback vertex set}…
We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of…
To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a…
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…
In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination,…
For a regular tournament $T$ of order $n,$ denote by $c_{8}(T)$ the number of cycles of length $8$ in $T.$ Let $DR_{n}$ be a doubly-regular tournament of order $n\equiv 3\mod4$ (so, the out-sets and in-sets of its vertices are also regular…
A square of a path on $k$ vertices is a directed path $x_1\ldots x_k$, where $x_i$ is directed to $x_{i+2}$, for every $i\in \{1,\ldots, k-1\}$. Recently, Yuster showed that any tournament on $n$ vertices contains a square of a path of…