On a problem of E. Prisner concerning the biclique operator

Bohdan Zelinka

Displaying similar documents to “On a problem of E. Prisner concerning the biclique operator”

Signed domination and signed domatic numbers of digraphs

Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If $\sum_{x \in N ¯ [v]} f (x) \geq 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number $γ_{S} (D)$ of D. A set $f ₁, f ₂, . . ., f_{d}$ of signed dominating functions on D with the property that...

The classification of finite groups by using iteration digraphs

Uzma Ahmad, Muqadas Moeen (2016)

Czechoslovak Mathematical Journal

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A digraph is associated with a finite group by utilizing the power map $f : G \to G$ defined by $f (x) = x^{k}$ for all $x \in G$ , where $k$ is a fixed natural number. It is denoted by $γ_{G} (n, k)$ . In this paper, the generalized quaternion and $2$ -groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a $2$ -group are determined for a $2$ -group to be a generalized quaternion group. Further, the classification of two generated $2$ -groups as abelian...

Self-diclique circulant digraphs

Marietjie Frick, Bernardo Llano, Rita Zuazua (2015)

Mathematica Bohemica

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We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let $D = (V, A)$ be a reflexive digraph (or network). Consider $X$ and $Y$ (not necessarily disjoint) nonempty subsets of vertices (or nodes) of $D$ . A disimplex $K (X, Y)$ of $D$ is...

On the heterochromatic number of circulant digraphs

Hortensia Galeana-Sánchez, Víctor Neumann-Lara (2004)

Discussiones Mathematicae Graph Theory

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The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let $D_{n, s}$ be the oriented graph such that $V (D_{n, s})$ is the set of integers mod 2n+1 and $A (D_{n, s}) = (i, j) : j - i \in 1, 2, . . ., n ∖ s . .$ In this paper we prove that $h c (D_{n, s}) \leq 5$ for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].

The total {k}-domatic number of digraphs

Seyed Mahmoud Sheikholeslami, Lutz Volkmann (2012)

Discussiones Mathematicae Graph Theory

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For a positive integer k, a total k-dominating function of a digraph D is a function f from the vertex set V(D) to the set 0,1,2, ...,k such that for any vertex v ∈ V(D), the condition $\sum_{u \in N^{-} (v)} f (u) \geq k$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set $f ₁, f ₂, . . ., f_{d}$ of total k-dominating functions of D with the property that $\sum_{i = 1}^{d} f_{i} (v) \leq k$ for each v ∈ V(D), is called a total k-dominating family (of functions) on D. The maximum number of functions in a total k-dominating family on D is...

On short cycles in triangle-free oriented graphs

Yurong Ji, Shufei Wu, Hui Song (2018)

Czechoslovak Mathematical Journal

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An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on $n$ vertices with minimum outdegree $d$ contains a directed cycle of length at most $⌈ n / d ⌉$ . In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that $α_{0}$ is the smallest real such that every $n$ -vertex digraph with minimum outdegree at least $α_{0} n$ contains a directed triangle. Let $ϵ < (3 - 2 α_{0}) / (4 - 2 α_{0})$ be a positive real. We show that if $D$ is an oriented graph...

Majority choosability of 1-planar digraph

Weihao Xia, Jihui Wang, Jiansheng Cai (2023)

Czechoslovak Mathematical Journal

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A majority coloring of a digraph $D$ with $k$ colors is an assignment $π : V (D) \to {1, 2, \dots, k}$ such that for every $v \in V (D)$ we have $π (w) = π (v)$ for at most half of all out-neighbors $w \in N^{+} (v)$ . A digraph $D$ is majority $k$ -choosable if for any assignment of lists of colors of size $k$ to the vertices, there is a majority coloring of $D$ from these lists. We prove that if $U (D)$ is a 1-planar graph without a 4-cycle, then $D$ is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable.

Iterated arc graphs

Danny Rorabaugh, Claude Tardif, David Wehlau, Imed Zaguia (2018)

Commentationes Mathematicae Universitatis Carolinae

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The arc graph $δ (G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $δ (G)$ join consecutive arcs of $G$ . In 1981, S. Poljak and V. Rödl characterized the chromatic number of $δ (G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number...

A note on a conjecture on niche hypergraphs

Pawaton Kaemawichanurat, Thiradet Jiarasuksakun (2019)

Czechoslovak Mathematical Journal

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For a digraph $D$ , the niche hypergraph $N ℋ (D)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges $E (N ℋ (D)) = {e \subseteq V (D) : | e | \geq 2$ and there exists a vertex $v$ such that $e = N_{D}^{-} (v)$ or $e = N_{D}^{+} (v)}$ . A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $ℋ$ , the niche number $\hat{n} (ℋ)$ is the smallest integer such that $ℋ$ together with $\hat{n} (ℋ)$ isolated vertices is the niche hypergraph of an acyclic digraph. C. Garske, M. Sonntag and H. M. Teichert (2016) conjectured that for a linear...