17 necessary and sufficient conditions for the primality of Fermat numbers
'3in1' enhanced: three squared ways to '3in1' GRAPHS
3-transitive digraphs
Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u,v,w,x) of length 3 in D implies the existence of the arc (u,x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g., to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs with...
4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs
Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v,w ∈ V (D), (u, v), (v,w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a long time, and...
(K − 1)-Kernels In Strong K-Transitive Digraphs
Let D = (V (D),A(D)) be a digraph and k ≥ 2 be an integer. A subset N of V (D) is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v) ≥ k; it is l-absorbent if for every u ∈ V (D) − N, there exists v ∈ N such that d(u, v) ≤ l. A (k, l)-kernel of D is a k-independent and l-absorbent subset of V (D). A k-kernel is a (k, k − 1)-kernel. A digraph D is k-transitive if for any path x0x1 ・ ・ ・ xk of length k, x0 dominates xk. Hernández-Cruz [3-transitive digraphs, Discuss. Math. Graph...
(k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings
Line digraphs can be obtained by sequences of state splittings, a particular kind of operation widely used in symbolic dynamics [12]. Properties of line digraphs inherited from the source have been studied, for instance in [7] Harminc showed that the cardinalities of the sets of kernels and solutions (kernel's dual definition) of a digraph and its line digraph coincide. We extend this for (k,l)-kernels in the context of state splittings and also look at (k,l)-semikernels, k-Grundy functions and...
γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs
We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ 0, 1, . . . , n there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices...
γ-Cycles In Arc-Colored Digraphs
We call a digraph D an m-colored digraph if the arcs of D are colored with m colors. A directed path (or a directed cycle) is called monochromatic if all of its arcs are colored alike. A subdigraph H in D is called rainbow if all of its arcs have different colors. A set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the two following conditions: for every pair of different vertices u, v ∈ N there is no monochromatic path in D between them, and for every vertex x ∈ V...
Информационные графы.
Комбинаторная характеристика конечных базисоов чистых свободных полугрупп
Об одном соотношении в конечных топологиях
Обобщенное соответствие Робинсона-Шенстеда-Кнута
Представление поликолчанов ручного типа
Свободные произведения сеток и свободные симметризаторы графов
Суммы степеней бинарных отношений