Nonstandard digraphs and transfinite digraphs have been defined and examined in two prior technical reports. The present work examines digraphs that are both nonstandard and transfinite. This requires a combination in certain ways of the techniques used in the prior two works. We first construct herein nonstandard transfinite digraphs for all the naturalnumber ranks. Then, a special kind of non...

Greene's Theorem states that the maximum cardinality of an optimal k-path in a poset is equal to the minimum k-norm of a k-optimal coloring. This result was extended to all acyclic digraphs, and is conjectured to hold for general digraphs. We prove the result for general digraphs in which an optimal k-path contains a path of cardinality one. This implies the validity of the conjecture for all b...

Gossiping is a problem of information dissemination described in a group of individuals connected by a communication network, where every individual in the network knows a unique item of information and needs to communicate it to everyone else. This communication pattern nds its main applications in the eld of interconnection networks for parallel and distributed architecture. Previous papers h...

The dichromatic number ~ χ(D) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the li...

In this paper, a colouring game and two versions of marking games (the weak and the strong) on digraphs are studied. We introduce the weak game chromatic number χwg(D) and the weak game colouring number wgcol(D) of digraphs D. It is proved that if D is an oriented planar graph, then χwg(D) 6 wgcol(D) 6 9, and if D is an oriented outerplanar graph, then χwg(D) 6 wgcol(D) 6 4. Then we study the s...

for a finite field $mathbb{f}_q$, the bivariate skew polynomial ring $mathbb{f}_q[x,y;rho,theta]$ has been used to study codes cite{xh}. in this paper, we give some characterizations of the ring $r[x,y;rho,theta]$, where $r$ is a commutative ring. we investigate 2-d skew $(lambda_1,lambda_2)$-constacyclic codes in the ring $r[x,y;rho,theta]/langle x^l-lambda_1,y^s-lambda_2rangle_{mathit{l}}.$ a...

The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection digraph from a given digraph.

In this paper, we study the kth upper bases of primitive non-powerful signed digraphs. A bound on the kth upper bases of all primitive non-powerful signed digraphs is obtained, and the equality case of the bound is characterized. We also show that there exists “gap” in the kth upper base set of primitive non-powerful signed digraphs. AMS classification: 05C20, 05C50, 15A48

The Klee-Quaife problem is finding the minimum order μ(d, c, v) of the (d, c, v) graph, which is a c-vertex connected v-regular graph with diameter d. Many authors contributed finding μ(d, c, v) and they also enumerated and classified the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, ...

sponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their Hasse digraphs called KoDAGs. KoDAGs are special orderable Directed Acyclic Graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. KoDAGs appear to be distinguished family of Ferrers digraphs which are natural join of a corresponding ordering ch...