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    Wyświetlanie 1-15 z 15
    Tytuł:
    On the b -Domatic Number of Graphs
    Autorzy:
    Benatallah, Mohammed
    Ikhlef-Eschouf, Noureddine
    Mihoubi, Miloud
    Powiązania:
    https://bibliotekanauki.pl/articles/31343453.pdf
    Data publikacji:
    2019-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domatic partition
    domatic number
    b-domatic partition
    b-domatic number
    Opis:
    A set of vertices S in a graph G = (V, E) is a dominating set if every vertex not in S is adjacent to at least one vertex in S. A domatic partition of graph G is a partition of its vertex-set V into dominating sets. A domatic partition P of G is called b-domatic if no larger domatic partition of G can be obtained from P by transferring some vertices of some classes of P to form a new class. The minimum cardinality of a b-domatic partition of G is called the b-domatic number and is denoted by bd(G). In this paper, we explain some properties of b-domatic partitions, and we determine the b-domatic number of some families of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 2; 313-324
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Roman Domatic Problem in Graphs and Digraphs: A Survey
    Autorzy:
    Chellali, Mustapha
    Rad, Nader Jafari
    Sheikholeslami, Seyed Mahmoud
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/32304148.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Roman domination
    domatic
    Opis:
    In this paper, we survey results on the Roman domatic number and its variants in both graphs and digraphs. This fifth survey completes our works on Roman domination and its variations published in two book chapters and two other surveys.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 861-891
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The b-Domatic Number of a Graph
    Autorzy:
    Favaron, Odile
    Powiązania:
    https://bibliotekanauki.pl/articles/30098013.pdf
    Data publikacji:
    2013-09-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domatic number
    adomatic number
    b-domatic number
    bchromatic number
    idomatic number
    partition
    Opis:
    Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic and adomatic numbers $d(G)$ and $ad(G)$. We introduce the b-domatic number $bd(G)$ as the counterpart of the b-chromatic number by giving an alternative definition of the maximality of a partition into dominating sets. We initiate the study of $bd(G)$ by giving some properties and examples.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 4; 747-757
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Almost Injective Colorings
    Autorzy:
    Goddard, Wayne
    Melville, Robert
    Xu, Honghai
    Powiązania:
    https://bibliotekanauki.pl/articles/31343577.pdf
    Data publikacji:
    2019-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    coloring
    injective
    closed neighborhood
    domatic
    Opis:
    We define an almost-injective coloring as a coloring of the vertices of a graph such that every closed neighborhood has exactly one duplicate. That is, every vertex has either exactly one neighbor with the same color as it, or exactly two neighbors of the same color. We present results with regards to the existence of such a coloring and also the maximum (minimum) number of colors for various graph classes such as complete k-partite graphs, trees, and Cartesian product graphs. In particular, we give a characterization of trees that have an almost-injective coloring. For such trees, we show that the minimum number of colors equals the maximum degree, and we also provide a polynomial-time algorithm for computing the maximum number of colors, even though these questions are NP-hard for general graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 1; 225-239
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Cubic Graphs with Total Domatic Number at Least Two
    Autorzy:
    Akbari, Saieed
    Motiei, Mohammad
    Mozaffari, Sahand
    Yazdanbod, Sina
    Powiązania:
    https://bibliotekanauki.pl/articles/31342441.pdf
    Data publikacji:
    2018-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total domination
    total domatic number
    coupon coloring
    Opis:
    Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 1; 75-82
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Double Roman Domatic Number of a Digraph
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31348166.pdf
    Data publikacji:
    2020-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    digraph
    double Roman domination
    double Roman domatic number
    Opis:
    A double Roman dominating function on a digraph $D$ with vertex set $V(D)$ is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function $f : V (D) → {0, 1, 2, 3}$ having the property that if $f(v) = 0$, then the vertex $v$ must have at least two in-neighbors assigned 2 under $f$ or one in-neighbor w with $f(w) = 3$, and $if f(v) = 1$, then the vertex v must have at least one in-neighbor $u$ with $f(u) ≥ 2$. A set ${f_1, f_2, . . ., f_d}$ of distinct double Roman dominating functions on $D$ with the property that $∑_{i=1}^df_i(v)≤3$ for each $v ∈ V (D)$ is called a double Roman dominating family (of functions) on $D$. The maximum number of functions in a double Roman dominating family on $D$ is the double Roman domatic number of $D$, denoted by $d_{dR}(D)$. We initiate the study of the double Roman domatic number, and we present different sharp bounds on $d_{dR}(D)$. In addition, we determine the double Roman domatic number of some classes of digraphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 4; 995-1004
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Tree domatic number in graphs
    Autorzy:
    Chen, X. G.
    Powiązania:
    https://bibliotekanauki.pl/articles/255594.pdf
    Data publikacji:
    2007
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    tree domatic number
    regular graph
    planar graph
    Cartesian product
    Opis:
    A dominating set S in a graph G is a tree dominating set of G if the subgraph induced by S is a tree. The tree domatic number of G is the maximum number of pairwise disjoint tree dominating sets in V(G). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most 4 and give a characterization of planar graphs with the tree domatic number 3.
    Źródło:
    Opuscula Mathematica; 2007, 27, 1; 5-11
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Partitioning a graph into a dominating set, a total dominating set, and something else
    Autorzy:
    Henning, Michael
    Löwenstein, Christian
    Rautenbach, Dieter
    Powiązania:
    https://bibliotekanauki.pl/articles/744065.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination
    total domination
    domatic number
    vertex partition
    Petersen graph
    Opis:
    A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 4; 563-574
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The k-rainbow domatic number of a graph
    Autorzy:
    Sheikholeslami, Seyyed
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/743715.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    k-rainbow dominating function
    k-rainbow domination number
    k-rainbow domatic number
    Opis:
    For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2, ...,k} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃_{u ∈ N(v)}f(u) = {1,2, ...,k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set ${f₁,f₂, ...,f_d}$ of k-rainbow dominating functions on G with the property that $∑_{i = 1}^d |f_i(v)| ≤ k$ for each v ∈ V(G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by $d_{rk}(G)$. Note that $d_{r1}(G)$ is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for $d_{rk}(G)$. Many of the known bounds of d(G) are immediate consequences of our results.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 1; 129-140
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Signed domination and signed domatic numbers of digraphs
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/743935.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    digraph
    oriented graph
    signed dominating function
    signed domination number
    signed domatic number
    Opis:
    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 $∑_{x ∈ N¯[v]}f(x) ≥ 1$ 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 ${f₁,f₂,...,f_d}$ of signed dominating functions on D with the property that $∑_{i = 1}^d f_i(x) ≤ 1$ for each x ∈ V(D), is called a signed dominating family (of functions) on D. The maximum number of functions in a signed dominating family on D is the signed domatic number of D, denoted by $d_S(D)$. In this work we show that $4-n ≤ γ_S(D) ≤ n$ for each digraph D of order n ≥ 2, and we characterize the digraphs attending the lower bound as well as the upper bound. Furthermore, we prove that $γ_S(D) + d_S(D) ≤ n + 1$ for any digraph D of order n, and we characterize the digraphs D with $γ_S(D) + d_S(D) = n + 1$. Some of our theorems imply well-known results on the signed domination number of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 3; 415-427
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Neighbourhood total domination in graphs
    Autorzy:
    Arumugam, S.
    Sivagnanam, C.
    Powiązania:
    https://bibliotekanauki.pl/articles/254824.pdf
    Data publikacji:
    2011
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    neighbourhood total domination
    total domination
    connected domination
    paired domination
    neighbourhood total domatic number
    Opis:
    Let G = (V, E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraph ⟨ N(S) ⟩ has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by ϒnt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G). In this paper we initiate a study of these parameters.
    Źródło:
    Opuscula Mathematica; 2011, 31, 4; 519-531
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The total {k}-domatic number of digraphs
    Autorzy:
    Sheikholeslami, Seyed
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/743233.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    digraph
    total {k}-dominating function
    total {k}-domination number
    total {k}-domatic number
    Opis:
    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 $∑_{u ∈ N^{ -}(v)}f(u) ≥ k$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set ${f₁,f₂, ...,f_d}$ of total {k}-dominating functions of D with the property that $∑_{i = 1}^d f_i(v) ≤ k$ 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 the total {k}-domatic number of D, denoted by $dₜ^{{k}}(D)$. Note that $dₜ^{{1}}(D)$ is the classic total domatic number $dₜ(D)$. In this paper we initiate the study of the total {k}-domatic number in digraphs, and we present some bounds for $dₜ^{{k}}(D)$. Some of our results are extensions of well-know properties of the total domatic number of digraphs and the total {k}-domatic number of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 3; 461-471
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Signed star (k, k)-domatic number of a graph
    Autorzy:
    Sheikholeslami, S. M.
    Volkmann, L.
    Powiązania:
    https://bibliotekanauki.pl/articles/254927.pdf
    Data publikacji:
    2014
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    signed star (k, k)-domatic number
    signed star domatic number
    signed star k-dominating function
    signed star dominating function
    signed star k-domination number
    signed star domination number
    regular graphs
    Opis:
    Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.
    Źródło:
    Opuscula Mathematica; 2014, 34, 3; 609-620
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Upper Bounds on the Signed Total (k, k)-Domatic Number of Graphs
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31339301.pdf
    Data publikacji:
    2015-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    signed total (k
    k)-domatic number
    signed total k-dominating function
    signed total k-domination number
    regular graphs
    Opis:
    Let $G$ be a graph with vertex set $V (G)$, and let $ f : V (G) \rightarrow {−1, 1}$ be a two-valued function. If $ k \geq 1$ is an integer and \( \sum_{ x \in N(v)} f(x) \geq k \) for each $ v \in V (G) $, where $N(v)$ is the neighborhood of $v$, then $f$ is a signed total $k$-dominating function on $G$. A set ${f_1, f_2, . . ., f_d}$ of distinct signed total k-dominating functions on $G$ with the property that \( \sum_{i=1}^d f_i(x) \leq k \) for each $ x \in V (G)$, is called a signed total ($k$, $k$)-dominating family (of functions) on $G$. The maximum number of functions in a signed total ($k$, $k$)-dominating family on $G$ is the signed total ($k$, $k$)-domatic number of $G$. In this article we mainly present upper bounds on the signed total ($k$, $k$)- domatic number, in particular for regular graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 4; 641-650
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Signed Total Roman k-Domatic Number Of A Graph
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31341581.pdf
    Data publikacji:
    2017-11-27
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    signed total Roman k-dominating function
    signed total Roman k-domination number
    signed total Roman k-domatic number
    Opis:
    Let $ k \ge 1 $ be an integer. A signed total Roman $k$-dominating function on a graph $G$ is a function $ f : V (G) \rightarrow {−1, 1, 2} $ such that $ \Sigma_{ u \in N(v) } f(u) \ge k $ for every $ v \in V (G) $, where $ N(v) $ is the neighborhood of $ v $, and every vertex $ u \in V (G) $ for which $ f(u) = −1 $ is adjacent to at least one vertex w for which $ f(w) = 2 $. A set $ { f_1, f_2, . . ., f_d} $ of distinct signed total Roman $k$-dominating functions on $G$ with the property that $ \Sigma_{i=1}^d f_i(v) \le k $ for each $ v \in V (G) $, is called a signed total Roman $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed total Roman $k$-dominating family on $G$ is the signed total Roman $k$-domatic number of $G$, denoted by $ d_{stR}^k (G) $. In this paper we initiate the study of signed total Roman $k$-domatic numbers in graphs, and we present sharp bounds for $ d_{stR}^k (G) $. In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman $k$-domatic number of some graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 4; 1027-1038
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
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