Related papers: Directed cycles with zero weight in $\mathbb{Z}_p^…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

Let $D=(V,A)$ be a digraph of order $n$ and let $W$ be any subset of $V$. We define the minimum semi-degree of $W$ in $D$ to be $\delta^0(W)=\mbox{min}\{\delta^+(W),\delta^-(W)\}$, where $\delta^+(W)$ is the minimum out-degree of $W$ in $D$…

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix…

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$…

A well-known result of Tutte says that if Gamma is an Abelian group and G is a graph having a nowhere-zero Gamma-flow, then G has a nowhere-zero Gamma'-flow for each Abelian group Gamma' whose order is at least the order of Gamma. Jaeger,…

A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$…

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$ contains a directed Hamilton cycle. In this paper we extend this theorem to a random…

An antidirected cycle in a digraph $G$ is a subdigraph whose underlying graph is a cycle, and in which no two consecutive edges form a directed path in $G$. Let $\sigma_{+-}(G)$ be the minimum value of $d^+(x)+d^-(y)$ over all pairs of…

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each…

Call a graph $G$ zero-forcing for a finite abelian group $\mathcal{G}$ if for every $\ell : V(G) \to \mathcal{G}$ there is a connected $A \subseteq V(G)$ with $\sum_{a \in A} \ell(a) = 0$. The problem we pose here is to characterise the…

Let p be a prime, let G be a p-valuable, abelian-by-procyclic group, and let k be a field of characteristic p. We will prove that all faithful prime ideals of the completed group algebra kG are controlled by the centre of G, and a complete…

Let $f(N)$ denote the least integer $k$ such that, if $G$ is an abelian group of order $N$ and $A \subseteq G$ is a uniformly random $k$-element subset, then with probability at least $\tfrac12$ the subset-sum set $\{ \sum_{x \in S} x : S…

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of…

A graph is called claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph $K_{1, 3}$. The undirected power graph of a group $G$ has vertices the elements of $G$, with an edge between $g_1$ and $g_2$ if one of…

Given any digraph $D$ on $n$ vertices, let $\mathcal{P}(D)$ be the family of all directed paths in $D$, and let $H$ be a digraph with the arc set $A(H)=\{a_1, \ldots, a_k\}$. The digraph $D$ is called arbitrary Hamiltonian $H$-linked if for…

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded…

We show that for every positive integer $k$, any tournament with minimum out-degree at least $(2+o(1))k^2$ contains a subdivision of the complete directed graph on $k$ vertices, which is best possible up to a factor of $8$. This may be…

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of…