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    Tytuł:
    Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs
    Autorzy:
    Jiang, Hui
    Li, Xueliang
    Zhang, Yingying
    Powiązania:
    https://bibliotekanauki.pl/articles/31343240.pdf
    Data publikacji:
    2019-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total-colored graph
    total monochromatic connection
    Erdős- Gallai-type problem
    Opis:
    A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. In this paper, we study two kinds of Erdős-Gallai-type problems for tmc(G) and completely solve them.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 4; 775-785
    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:
    More on the Rainbow Disconnection in Graphs
    Autorzy:
    Bai, Xuqing
    Chang, Renying
    Huang, Zhong
    Li, Xueliang
    Powiązania:
    https://bibliotekanauki.pl/articles/32222544.pdf
    Data publikacji:
    2022-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge-coloring
    edge-connectivity
    rainbow disconnection coloring (number)
    Erdős-Gallai type problem
    Nordhaus-Gaddum type bounds
    complexity
    NP-hard (complete)
    Opis:
    Let G be a nontrivial edge-colored connected graph. An edge-cut R of G is called a rainbow-cut if no two of its edges are colored the same. An edge-colored graph G is rainbow disconnected if for every two vertices u and v of G, there exists a u-v-rainbow-cut separating them. For a connected graph G, the rainbow disconnection number of G, denoted by rd(G), is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. In this paper, we first determine the maximum size of a connected graph G of order n with rd(G) = k for any given integers k and n with 1 ≤ k ≤ n − 1, which solves a conjecture posed only for n odd in [G. Chartrand, S. Devereaux, T.W. Haynes, S.T. Hedetniemi and P. Zhang, Rainbow disconnection in graphs, Discuss. Math. Graph Theory 38 (2018) 1007–1021]. From this result and a result in their paper, we obtain Erdős-Gallai type results for rd(G). Secondly, we discuss bounds on rd(G) for complete multipartite graphs, critical graphs with respect to the chromatic number, minimal graphs with respect to the chromatic index, and regular graphs, and we also give the values of rd(G) for several special graphs. Thirdly, we get Nordhaus-Gaddum type bounds for rd(G), and examples are given to show that the upper and lower bounds are sharp. Finally, we show that for a connected graph G, to compute rd(G) is NP-hard. In particular, we show that it is already NP-complete to decide if rd(G) = 3 for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph G it is NP-complete to decide whether G is rainbow disconnected.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 4; 1185-1204
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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