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    Wyświetlanie 1-2 z 2
    Tytuł:
    Color-bounded hypergraphs, V: host graphs and subdivisions
    Autorzy:
    Bujtás, Csilla
    Tuza, Zsolt
    Voloshin, Vitaly
    Powiązania:
    https://bibliotekanauki.pl/articles/743863.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    mixed hypergraph
    color-bounded hypergraph
    vertex coloring
    arboreal hypergraph
    hypertree
    feasible set
    host graph
    edge subdivision
    Opis:
    A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = {E₁,...,Eₘ}, together with integers $s_i$ and $t_i$ satisfying $1 ≤ s_i ≤ t_i ≤ |E_i|$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_i$ satisfies $s_i ≤ |φ(E_i)| ≤ t_i$. The hypergraph ℋ is colorable if it admits at least one proper coloring.
    We consider hypergraphs ℋ over a "host graph", that means a graph G on the same vertex set X as ℋ, such that each $E_i$ induces a connected subgraph in G. In the current setting we fix a graph or multigraph G₀, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G₀.
    The colorability problem is known to be NP-complete in general, and also when restricted to 3-uniform "mixed hypergraphs", i.e., color-bounded hypergraphs in which $|E_i| = 3$ and $1 ≤ s_i ≤ 2 ≤ t_i ≤ 3$ holds for all i ≤ m. We prove that for every fixed graph G₀ and natural number r, colorability is decidable in polynomial time over the class of r-uniform hypergraphs (and more generally of hypergraphs with $|E_i| ≤ r$ for all 1 ≤ i ≤ m) having a host graph G obtained from G₀ by edge subdivisions. Stronger bounds are derived for hypergraphs for which G₀ is a tree.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 2; 223-238
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Irreducible No-Hole L(2, 1)-Coloring of Edge-Multiplicity-Paths-Replacement Graph
    Autorzy:
    Mandal, Nibedita
    Panigrahi, Pratima
    Powiązania:
    https://bibliotekanauki.pl/articles/31342318.pdf
    Data publikacji:
    2018-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    L (2,1)-coloring
    no-hole coloring
    irreducible coloring
    subdivision graph
    edge-multiplicity-paths-replacement graph
    Opis:
    An L(2, 1)-coloring (or labeling) of a simple connected graph G is a mapping f : V (G) → Z+ ∪ {0} such that |f(u)−f(v)| ≥ 2 for all edges uv of G, and |f(u) − f(v)| ≥ 1 if u and v are at distance two in G. The span of an L(2, 1)-coloring f, denoted by span(f), of G is max{f(v) : v ∈ V (G)}. The span of G, denoted by λ(G), is the minimum span of all possible L(2, 1)-colorings of G. For an L(2, 1)-coloring f of a graph G with span k, an integer l is a hole in f if l ∈ (0, k) and there is no vertex v in G such that f(v) = l. An L(2, 1)-coloring is a no-hole coloring if there is no hole in it, and is an irreducible coloring if color of none of the vertices in the graph can be decreased and yield another L(2, 1)-coloring of the same graph. An irreducible no-hole coloring, in short inh-coloring, of G is an L(2, 1)-coloring of G which is both irreducible and no-hole. For an inh-colorable graph G, the inh-span of G, denoted by λinh(G), is defined as λinh(G) = min{span(f) : f is an inh-coloring of G. Given a function h : E(G) → ℕ − {1}, and a positive integer r ≥ 2, the edge-multiplicity-paths-replacement graph G(rPh) of G is the graph obtained by replacing every edge uv of G with r paths of length h(uv) each. In this paper we show that G(rPh) is inh-colorable except possibly the cases h(e) ≥ 2 with equality for at least one but not for all edges e and (i) Δ(G) = 2, r = 2 or (ii) Δ (G) ≥ 3, 2 ≤ r ≤ 4. We find the exact value of λinh(G(rPh)) in several cases and give upper bounds of the same in the remaining. Moreover, we find the value of λ(G(rPh)) in most of the cases which were left by Lü and Sun in [L(2, 1)-labelings of the edge-multiplicity-paths-replacement of a graph, J. Comb. Optim. 31 (2016) 396–404].
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 2; 525-552
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-2 z 2

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