Related papers: On cyclotomic nearly-doubly-regular tournaments
We show that if $T$ is a strongly $10^9k^6\log(2k)$-connected tournament, there exists a partition $A, B$ of $V(T)$ such that each of $T[A]$, $T[B]$ and $T[A,B]$ is strongly $k$-connected. This provides tournament analogues of two partition…
K\"uhn and Osthus conjectured in 2013 that regular tripartite tournaments are decomposable into Hamilton cycles. Somewhat surprisingly, Granet gave a simple counterexample to this conjecture almost 10 years later. In this paper, we show…
The incomplete round robin sports tournament format, where each team plays the same number of games but faces only a subset of the other teams, is becoming increasingly popular in both youth and professional competitions. In contrast to…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
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…
In his paper "Kings in Bipartite Hypertournaments" (Graphs $\&$ Combinatorics 35, 2019), Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.
A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding…
We study some problems pertaining to the tournament equilibrium set (TEQ for short). A tournament $H$ is a TEQ-retentive tournament if there is a tournament $T$ which has a minimal TEQ-retentive set $R$ such that $T[R]$ is isomorphic to…
We provide a detailed study of topological and combinatorial properties of sectionable tournaments. This class forms an inductively constructed family of tournaments grounded over simply disconnected tournaments, those tournaments whose…
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the…
We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov's…
Scott and Seymour conjectured the existence of a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\chi(G) \geqslant f(k)$ implies that $\chi(G[N_T^+(v)]) \geqslant k$…
A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number $S_n$ of score sequences on the…
We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is…
The Erd\H{o}s-Hajnal Conjecture states that for every $H$ there exists a constant $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least…
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…
Crucial to an Evolutionary Algorithm's performance is its selection scheme. We mathematically investigate the relation between polynomial rank and probabilistic tournament methods which are (respectively) generalisations of the popular…
We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the…
A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of verticesof $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels.The distinguishing number $D(G)$…
A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of…