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    Wyświetlanie 1-8 z 8
    Tytuł:
    Sharp Upper Bounds for Generalized Edge-Connectivity of Product Graphs
    Autorzy:
    Sun, Yuefang
    Powiązania:
    https://bibliotekanauki.pl/articles/31340760.pdf
    Data publikacji:
    2016-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    generalized edge-connectivity
    Cartesian product
    strong product
    lexicographic product
    Opis:
    The generalized $k$-connectivity $ \kappa_k (G) $ of a graph $G$ was introduced by Hager in 1985. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as $ \lambda k(G) = \text{min} \{ \lambda (S) : S \subseteq V (G) \text{ and } |S| = k \} $, where $ \lambda(S) $ denote the maximum number $ \mathcal{l} $ of pairwise edge-disjoint trees $ T_1, T_2, . . ., T_\mathcal{l} $ in $G$ such that $ S \subseteq V ( T_i ) $ for $ 1 \le i \le \mathcal{l} $. In this paper, we study the generalized edge- connectivity of product graphs and obtain sharp upper bounds for the generalized 3-edge-connectivity of Cartesian product graphs and strong product graphs. Among our results, some special cases are also discussed.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2016, 36, 4; 833-843
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A Sharp Lower Bound For The Generalized 3-Edge-Connectivity Of Strong Product Graphs
    Autorzy:
    Sun, Yuefang
    Powiązania:
    https://bibliotekanauki.pl/articles/31341589.pdf
    Data publikacji:
    2017-11-27
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    generalized connectivity
    generalized edge-connectivity
    strong product
    Opis:
    The generalized $k$-connectivity $ \kappa_k (G) $ of a graph $G$, mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as $ \lambda_k (G) = \text{min} \{ \lambda_G (S) | S \subseteq V (G) $ and $ |S| = k \} $, where $ \lambda_G (S) $ denote the maximum number $ \mathcal{l} $ of pairwise edge-disjoint trees $ T_1 $, $ T_2 $, . . ., $ T_\mathcal{l} $ in $G$ such that $S \subseteq V (T_i)$ for $ 1 \le i \le \mathcal{l} $. In this paper we get a sharp lower bound for the generalized 3-edge-connectivity of the strong product of any two connected graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 4; 975-988
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the Maximum and Minimum Sizes of a Graph with Given k-Connectivity
    Autorzy:
    Sun, Yuefang
    Powiązania:
    https://bibliotekanauki.pl/articles/31341709.pdf
    Data publikacji:
    2017-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    k -connectivity
    generalized connectivity
    connectivity
    Opis:
    The concept of k-connectivity κk(G), introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. For an integer k ≥ 2, the k-connectivity of a connected graph G with order n ≥ k is the smallest number of vertices whose removal from G produces a graph with at least k components or a graph with fewer than k vertices. In this paper, we get a sharp upper bound for the size of G with κk(G) = t, where 1 ≤ t ≤ n − k and k ≥ 3; moreover, the unique extremal graph is given. Based on this result, we get the exact values for the maximum size, denoted by g(n, k, t), of a connected graph G with order n and κk(G) = t. We also compute the exact values and bounds for another parameter f(n, k, t) which is defined as the minimum size of a connected graph G with order n and κk(G) = t, where 1 ≤ t ≤ n − k and k ≥ 3.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 3; 623-632
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Minimally Strong Subgraph (k, ℓ)-Arc-Connected Digraphs
    Autorzy:
    Sun, Yuefang
    Jin, Zemin
    Powiązania:
    https://bibliotekanauki.pl/articles/32304302.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    strong subgraph k -connectivity
    strong subgraph k -arc-connectivity
    subdigraph packing
    Opis:
    Let $D = (V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $ 2 \le k \le n$. A subdigraph $H$ of $D$ is called an $S$-strong subgraph if $H$ is strong and $ S \subseteq V (H) $. Two $S$-strong subgraphs $ D_1 $ and $ D_2 $ are said to be arc-disjoint if $ A(D_1) \cap A(D_2) = \emptyset $. Let $ \lambda_S (D) $ be the maximum number of arc-disjoint $S$-strong digraphs in $D$. The strong subgraph $k$-arc-connectivity is defined as $ \lambda_k (D) = \text{min} \{ \lambda_S (D) | S \subseteq V, |S| = k \} $. A digraph $ D = (V, A) $ is called minimally strong subgraph $ (k, \mathcal{l})$-arc-connected if $ \lambda_k (D) \ge \mathcal{l} $ but for any arc $ e \in A $, $ \lambda_k(D − e) \le \mathcal{l} − 1 $. Let \( \mathfrak{G}(n, k, \mathscr{l} ) \) be the set of all minimally strong subgraph $ (k, \mathcal{l} )$-arc-connected digraphs with order $n$. We define $ G(n, k, \mathcal{l} ) = $ \( \max \{ |A(D)| \ | D \in \mathfrak{G} (n, k, \mathcal{l} ) \} \) and $ g(n, k, \mathcal{l} ) = $ \( \min \{ |A(D)| \ | D \in \mathfrak{G}(n, k, \mathcal{l} ) \} \). In this paper, we study the minimally strong subgraph $ (k, \mathcal{l} ) $-arc-connected digraphs. We give a characterization of the minimally strong sub-graph $ (3, n − 2) $-arc-connected digraphs, and then give exact values and bounds for the functions $ g(n, k, \mathcal{l} )$ and $ G(n, k, \mathcal{l} ) $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 759-770
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Rainbow Connection Number of Graphs with Diameter 3
    Autorzy:
    Li, Hengzhe
    Li, Xueliang
    Sun, Yuefang
    Powiązania:
    https://bibliotekanauki.pl/articles/31342160.pdf
    Data publikacji:
    2017-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge-coloring
    rainbow path
    rainbow connection number
    diameter
    Opis:
    A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 1; 141-154
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Rainbow Total-Coloring of Complementary Graphs and Erdős-Gallai Type Problem For The Rainbow Total-Connection Number
    Autorzy:
    Sun, Yuefang
    Jin, Zemin
    Tu, Jianhua
    Powiązania:
    https://bibliotekanauki.pl/articles/31342242.pdf
    Data publikacji:
    2018-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Rainbow total-coloring
    rainbow total-connection number
    complementary graph
    Erdős-Gallai type problem
    Opis:
    A total-colored graph $G$ is rainbow total-connected if any two vertices of $G$ are connected by a path whose edges and internal vertices have distinct colors. The rainbow total-connection number, denoted by $ rtc(G) $, of a graph $G$ is the minimum number of colors needed to make $G$ rainbow total-connected. In this paper, we prove that $ rtc(G) $ can be bounded by a constant 7 if the following three cases are excluded: $ diam( \overline{G} ) = 2 $, $ diam( \overline{G} ) = 3 $, $ \overline{G} $ contains exactly two connected components and one of them is a trivial graph. An example is given to show that this bound is best possible. We also study Erdős-Gallai type problem for the rainbow total-connection number, and compute the lower bounds and precise values for the function $ f(n, k) $, where $ f(n, k) $ is the minimum value satisfying the following property: if $ |E(G)| \ge f(n, k) $, then $ rtc(G) \le k $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 4; 1023-1036
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Two Generalized Connectivities of Graphs
    Autorzy:
    Sun, Yuefang
    Li, Fengwei
    Jin, Zemin
    Powiązania:
    https://bibliotekanauki.pl/articles/31342427.pdf
    Data publikacji:
    2018-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    k -connectivity
    pendant tree-connectivity
    Cartesian product
    Cayley graph
    Opis:
    The concept of generalized $k$-connectivity $ \kappa_k (G) $, mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity $ \tau_k (G) $ was also introduced by Hager in 1985, which is a specialization of generalized $k$-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph $G$, named $k$-connectivity $ \kappa_k^' (G) $, introduced by Chartrand et al. in 1984, is a generalization of the cut-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of $ \kappa_k^' (G) $ and $ \tau_k(G) $ by showing that for a connected graph $G$ of order $n$, if $ \kappa_k^' (G) \ne n − k + 1 $ where $ k \ge 3 $, then $ 1 \le \kappa_k^' (G) − \tau_k (G) \le n − k $; otherwise, $ 1 \le κ_k^' (G) − \tau_k(G) \le n − k + 1 $. Moreover, all of these bounds are sharp. We get a sharp upper bound for the 3-connectivity of the Cartesian product of any two connected graphs with orders at least 5. Especially, the exact values for some special cases are determined. Among our results, we also study the pendant tree-connectivity of Cayley graphs on Abelian groups of small degrees and obtain the exact values for $ \tau_k(G) $, where $G$ is a cubic or 4-regular Cayley graph on Abelian groups, $ 3 \le k \le n $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 1; 245-261
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Minimum Size of a Graph with Given Tree Connectivity
    Autorzy:
    Sun, Yuefang
    Sheng, Bin
    Jin, Zemin
    Powiązania:
    https://bibliotekanauki.pl/articles/32083875.pdf
    Data publikacji:
    2021-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    generalized connectivity
    tree connectivity
    eneralized k-connectivity
    generalized k-edge-connectivity
    packing
    Opis:
    For a graph $G = (V, E)$ and a set $S ⊆ V$ of at least two vertices, an $S$-tree is a such subgraph $T$ of $G$ that is a tree with $S ⊆ V(T)$. Two $S$-trees $T_1$ and $T_2$ are said to be internally disjoint if $E(T_1) ∩ E(T_2) = ∅$ and $V(T_1) ∩ V(T_2) = S$, and edge-disjoint if $E(T_1) ∩ E(T_2) = ∅$. The generalized local connectivity $κ_G(S)$ (generalized local edge-connectivity $λ_G(S)$, respectively) is the maximum number of internally disjoint (edge-disjoint, respectively) $S$-trees in $G$. For an integer $k$ with $2 ≤ k ≤ n$, the generalized $k$-connectivity (generalized $k$-edge-connectivity, respectively) is defined as $κ_k(G) = min{κ_G(S) | S ⊆ V(G), |S| = k} (λ_k(G) = min{λ_G(S) | S ⊆ V(G), |S| = k}$, respectively). Let $f(n, k, t)$ ($g(n, k, t)$, respectively) be the minimum size of a connected graph $G$ with order $n$ and $κ_k(G) = t$ ($λ_k(G) = t$, respectively), where $3 ≤ k ≤ n$ and \(1≤t≤n-⌈\frac{k}{2}⌉\). For general $k$ and $t$, Li and Mao obtained a lower bound for $g(n, k, t)$ which is tight for the case $k = 3$. We show that the bound also holds for $f(n, k, t)$ and is tight for the case $k = 3$. When t is general, we obtain upper bounds of both $f(n, k, t)$ and $g(n, k, t)$ for $k ∈ {3, 4, 5}$, and all of these bounds can be attained. When $k$ is general, we get an upper bound of $g(n, k, t)$ for $t ∈ {1, 2, 3, 4}$ and an upper bound of $f(n, k, t)$ for $t ∈ {1, 2, 3}$. Moreover, both bounds can be attained.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 2; 409-425
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-8 z 8

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