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    Wyświetlanie 1-6 z 6
    Tytuł:
    Achromatic Numbers for Circulant Graphs and Digraphs
    Autorzy:
    Araujo-Pardo, Gabriela
    Montellano-Ballesteros, Juan José
    Olsen, Mika
    Rubio-Montiel, Christian
    Powiązania:
    https://bibliotekanauki.pl/articles/32222712.pdf
    Data publikacji:
    2021-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    circulant graphs
    complete colorings
    achromatic number
    achromatic index
    Opis:
    In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $ \alpha (C_{16q^2 + 20q + 7}(1, 2)) = 8q + 5 $, and for the diachromatic number we state that $ dac( \vec{C}_{32q^2 + 24q + 5} (1, 2)) = 8q + 3$. In general, we give the lower bounds $ \alpha(C_{4q^2 + aq+1} (1, a)) \ge 4q + 1$ and $ dac( \vec{C}_{8q^2+2(a+4)q+a+3} (1, a)) \ge 4q + 3 $ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1} $ for graphs and $ \mathbb{Z}_{4q+3} $ for digraphs, and the equality is attained, in both cases, for $a = 3$. Finally, we determine the achromatic index for circulant graphs of $ q^2 +q + 1 $ vertices when the projective cyclic plane of odd order $q$ exists.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 3; 713-724
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Open Locating-Dominating Sets in Circulant Graphs
    Autorzy:
    Givens, Robin M.
    Yu, Gexin
    Kincaid, Rex K.
    Powiązania:
    https://bibliotekanauki.pl/articles/32361753.pdf
    Data publikacji:
    2022-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    open locating-dominating sets
    circulant graphs
    Hall’s Matching Theorem
    mixed-weight open locating-dominating sets
    Opis:
    Location detection problems have been studied for a variety of applications including finding faults in multiprocessors, contaminants in public utilities, intruders in buildings and facilities, and for environmental monitoring using wireless sensor networks. In each of these applications, the system or structure can be modeled as a graph, and sensors placed strategically at a subset of vertices can locate and detect anomalies in the system. An open locating-dominating set (OLD-set) is a subset of vertices in a graph in which every vertex in the graph has a non-empty and unique set of neighbors in the subset. Sensors placed at OLD-set vertices can uniquely detect and locate disturbances in a system. These sensors can be expensive and, as a result, minimizing the size of the OLD-set is critical. Circulant graphs, a group of regular cyclic graphs, are often used to model parallel networks. We prove the optimal OLD-set size for a particular circulant graph using Hall’s Theorem. We also consider the mixed-weight OLD-set introduced in [R.M. Givens, R.K. Kincaid, W. Mao and G. Yu, Mixed-weight open locating-dominating sets, in: 2017 Annual Conference on Information Science and Systems, (IEEE, Baltimor, 2017) 1–6] which models a system with sensors of varying strengths. To model these systems, we place weights on the vertices in the graph, representing the strength of a sensor placed at the corresponding location in the system. We study particular mixed-weight OLD-sets in cycles, which behave similarly to OLD-sets in circulant graphs, and show the optimal mixed-weight OLD-set size using the discharging method.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 1; 47-62
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    L(2, 1)-Labeling of Circulant Graphs
    Autorzy:
    Mitra, Sarbari
    Bhoumik, Soumya
    Powiązania:
    https://bibliotekanauki.pl/articles/31343649.pdf
    Data publikacji:
    2019-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph coloring
    L(2
    1)-labeling
    circulants
    Opis:
    An $L(2, 1)$-labeling of a graph $ \Gamma $ is an assignment of non-negative integers to the vertices such that adjacent vertices receive labels that differ by at least 2, and those at a distance of two receive labels that differ by at least one. Let $ \lambda_2^1 (\Gamma) $ denote the least $ \lambda $ such that $ \Gamma $ admits an $ L(2, 1) $-labeling using labels from $ \{ 0, 1, . . ., \lambda \} $. A Cayley graph of group $G$ is called a circulant graph of order $n$, if $ G = \mathbb{Z}_n$. In this paper initially we investigate the upper bound for the span of the $L(2, 1)$-labeling for Cayley graphs on cyclic groups with “large” connection sets. Then we extend our observation and find the span of $L(2, 1)$-labeling for any circulants of order $n$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 1; 143-155
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The metric dimension of circulant graphs and their Cartesian products
    Autorzy:
    Chau, K.
    Gosselin, S.
    Powiązania:
    https://bibliotekanauki.pl/articles/255804.pdf
    Data publikacji:
    2017
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    metric dimension
    circulant graph
    Cartesian product
    Opis:
    Let G = (V, E) be a connected graph (or hypergraph) and let d(x,y) denote the distance between vertices x,y ∈V(G). A subset W ⊆V(G) is called a resolving set for G if for every pair ol distinct vertices x, y ∈ (G), there is w ∈W such that d(x,w) ≠d(y,w). The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β (G). The circulant graph Cn(l, 2,... , t) has vertex set {v0, v1 …, vn-1} and edges [formula] where 0 ≤ i ≤ n — 1 and 1 ≤j ≤ t and the indices are taken modulo [formula]. In this paper we determine the exact metric dimension olthe circulant graphs Cn(l, 2,... , t). extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrik (2016). In particular, we show that [formula] for large enough n, which implies that the metric dimension ol these circulants is completely determined by the congruence class ol n modulo 2t. We determine the exact value of β Cn (l, 2,.. . , i)) for n ≡ 2 mod 2t and n =≡ (t + 1) mod 2t and we give better bounds on the metric dimension ol these circulants for n ≡ 0 mod 2t and n ≡ 1 mod 2t. In addition, we bound the metric dimension ol Cartesian products ol circulant graphs.
    Źródło:
    Opuscula Mathematica; 2017, 37, 4; 509-534
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the Metric Dimension of Directed and Undirected Circulant Graphs
    Autorzy:
    Vetrík, Tomáš
    Powiązania:
    https://bibliotekanauki.pl/articles/31870010.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    metric dimension
    resolving set
    circulant graph
    distance
    Opis:
    The undirected circulant graph $C_n(±1, ±2, . . ., ±t)$ consists of vertices $v_0, v_1, . . ., v_{n−1}$ and undirected edges $v_iv_{i+j}$, where $0 ≤ i ≤ n − 1, 1 ≤ j ≤ t (2 ≤ t ≤ \frac{n}{2})$, and the directed circulant graph $C_n(1, t)$ consists of vertices $v_0, v_1, . . ., v_{n−1}$ and directed edges $v_iv_{i+1}, v_iv_{i+t}$, where $0 ≤ i ≤ n − 1 (2 ≤ t ≤ n−1)$, the indices are taken modulo $n$. Results on the metric dimension of undirected circulant graphs $C_n(±1, ±t)$ are available only for special values of $t$. We give a complete solution of this problem for directed graphs $C_n(1, t)$ for every $t ≥ 2$ if $n ≥ 2t^2$. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Appl. Math. Comput. 248 (2014) 47–54] presented a conjecture saying that dim $(C_n(±1, ±2, . . ., ±t)) = t + p − 1$ for $n = 2tk + t + p$, where $3 ≤ p ≤ t + 1$. We disprove it by showing that dim $(C_n(±1, ±2, . . ., ±t)) ≤ t + \frac{p+1}{2}$ for $n = 2tk + t + p$, where $t ≥ 4$ is even, $p$ is odd, $1 ≤ p ≤ t + 1$ and $k ≥ 1$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 67-76
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Erdős regular graphs of even degree
    Autorzy:
    Dobrynin, Andrey
    Mel'nikov, Leonid
    Pyatkin, Artem
    Powiązania:
    https://bibliotekanauki.pl/articles/743782.pdf
    Data publikacji:
    2007
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    vertex coloring
    4-critical graph
    circulant
    regular graph
    vertex connectivity
    Opis:
    In 1960, Dirac put forward the conjecture that r-connected 4-critical graphs exist for every r ≥ 3. In 1989, Erdös conjectured that for every r ≥ 3 there exist r-regular 4-critical graphs. A method for finding r-regular 4-critical graphs and the numbers of such graphs for r ≤ 10 have been reported in [6,7]. Results of a computer search for graphs of degree r = 12,14,16 are presented. All the graphs found are both r-regular and r-connected.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2007, 27, 2; 269-279
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-6 z 6

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