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    Wyświetlanie 1-6 z 6
    Tytuł:
    Coloring rectangular blocks in 3-space
    Autorzy:
    Magnant, Colton
    Martin, Daniel
    Powiązania:
    https://bibliotekanauki.pl/articles/743845.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    chromatic number
    channel assignment problem
    3 dimensional rectangular blocks
    Opis:
    If rooms in an office building are allowed to be any rectangular solid, how many colors does it take to paint any configuration of rooms so that no two rooms sharing a wall or ceiling/floor get the same color? In this work, we provide a new construction which shows this number can be arbitrarily large.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 1; 161-170
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A decomposition of Gallai multigraphs
    Autorzy:
    Halperin, Alexander
    Magnant, Colton
    Pula, Kyle
    Powiązania:
    https://bibliotekanauki.pl/articles/30148236.pdf
    Data publikacji:
    2014-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge coloring
    Gallai multigraph
    Opis:
    An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 2; 331-352
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Proper (Strong) Rainbow Connection of Graphs
    Autorzy:
    Jiang, Hui
    Li, Wenjing
    Li, Xueliang
    Magnant, Colton
    Powiązania:
    https://bibliotekanauki.pl/articles/32083886.pdf
    Data publikacji:
    2021-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    proper (strong) rainbow connection number
    Cartesian product
    chromatic index
    Opis:
    A path in an edge-colored graph $G$ is called a rainbow path if no two edges on the path have the same color. The graph $G$ is called rainbow connected if between every pair of distinct vertices of $G$, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of $G$ is proper. The proper rainbow connection number of $G$, denoted by $prc(G)$, is the minimum number of colors needed to properly color the edges of $G$ so that $G$ is rainbow connected. Similarly, the proper strong rainbow connection number of $G$, denoted by $psrc(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that for any two distinct vertices of $G$, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if \(G = K_{p1} \Box \dots \Box K_{pn}\), where $n≥ 1$, and $p_1, . . ., p_n>1$ are integers, then $prc(G) = psrc(G) = χ^′(G)$, where $χ^′(G)$ denotes the chromatic index of $G$. Finally, we investigate some suffcient conditions for a graph $G$ to satisfy $prc(G) = rc(G)$, and make some slightly positive progress by using a relation between $rc(G)$ and the girth of the graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 2; 469-479
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Generalized Rainbow Connection of Graphs and their Complements
    Autorzy:
    Li, Xueliang
    Magnant, Colton
    Wei, Meiqin
    Zhu, Xiaoyu
    Powiązania:
    https://bibliotekanauki.pl/articles/31342420.pdf
    Data publikacji:
    2018-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    ℓ-rainbow path
    ( k, ℓ)-rainbow connected
    ( k, ℓ)-rainbow connection number
    Opis:
    Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called $ \mathcal{l} $-rainbow if each subpath of length at most $ \mathcal{l} + 1 $ is rainbow. The graph $G$ is called $(k, \mathcal{l} )$-rainbow connected if there is an edge-coloring such that every pair of distinct vertices of $G$ is connected by $k$ pairwise internally vertex-disjoint $ \mathcal{l} $-rainbow paths in $G$. The minimum number of colors needed to make $G$ $(k, \mathcal{l})$-rainbow connected is called the $ (k, \mathcal{l} )$-rainbow connection number of $G$ and denoted by $ rc_{ k,\mathcal{l} } (G) $. In this paper, we first focus on the (1, 2)-rainbow connection number of $G$ depending on some constraints of $ \overline{G} $. Then, we characterize the graphs of order $n$ with (1, 2)-rainbow connection number $ n − 1 $ or $ n − 2 $. Using this result, we investigate the Nordhaus-Gaddum-Type problem of (1, 2)-rainbow connection number and prove that $ rc_{1,2}(G) + rc_{1,2}( \overlina{G} ) \le n + 2 $ for connected graphs $ G $ and $ \overline{G} $. The equality holds if and only if $G$ or $ \overline{G} $ is isomorphic to a double star.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 2; 371-384
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Chvátal-Erdös type theorems
    Autorzy:
    Faudree, Jill
    Faudree, Ralph
    Gould, Ronald
    Jacobson, Michael
    Magnant, Colton
    Powiązania:
    https://bibliotekanauki.pl/articles/744255.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Hamiltonian
    Hamiltonian-connected
    Chvátal-Erdös condition
    independence number
    Opis:
    The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-k)/(k+1), and δ(G) ≥ α(G)+k-2, then G is hamiltonian. It is shown that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ 4k²+1, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected. This result supports the conjecture that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected, and the conjecture is verified for k = 3 and 4.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 2; 245-256
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Proper Diameter of a Graph
    Autorzy:
    Coll, Vincent
    Hook, Jonelle
    Magnant, Colton
    McCready, Karen
    Ryan, Kathleen
    Powiązania:
    https://bibliotekanauki.pl/articles/32062335.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    diameter
    properly connected
    proper diameter
    Opis:
    A proper edge-coloring of a graph is a coloring in which adjacent edges receive distinct colors. A path is properly colored if consecutive edges have distinct colors, and an edge-colored graph is properly connected if there exists a properly colored path between every pair of vertices. In such a graph, we introduce the notion of the graph’s proper diameter—which is a function of both the graph and the coloring—and define it to be the maximum length of a shortest properly colored path between any two vertices in the graph. We consider various families of graphs to find bounds on the gap between the diameter and possible proper diameters, paying singular attention to 2-colorings.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 107-125
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-6 z 6

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