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    Wyświetlanie 1-6 z 6
    Tytuł:
    All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5
    Autorzy:
    Borodin, Oleg V.
    Ivanova, Anna O.
    Powiązania:
    https://bibliotekanauki.pl/articles/31342193.pdf
    Data publikacji:
    2017-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    3-polytope
    planar graph
    structure properties
    k -star
    Opis:
    Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s: there is a vertex of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P5s. Also, we give a tight description of stars with at least three rays in P5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P5s that might be useful in further research.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 1; 5-12
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Signed star (k, k)-domatic number of a graph
    Autorzy:
    Sheikholeslami, S. M.
    Volkmann, L.
    Powiązania:
    https://bibliotekanauki.pl/articles/254927.pdf
    Data publikacji:
    2014
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    signed star (k, k)-domatic number
    signed star domatic number
    signed star k-dominating function
    signed star dominating function
    signed star k-domination number
    signed star domination number
    regular graphs
    Opis:
    Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.
    Źródło:
    Opuscula Mathematica; 2014, 34, 3; 609-620
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Degree Sum Condition for the Existence of Spanning k-Trees in Star-Free Graphs
    Autorzy:
    Furuya, Michitaka
    Maezawa, Shun-ichi
    Matsubara, Ryota
    Matsuda, Haruhide
    Tsuchiya, Shoichi
    Yashima, Takamasa
    Powiązania:
    https://bibliotekanauki.pl/articles/32361756.pdf
    Data publikacji:
    2022-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    spanning tree
    k -tree
    star-free
    degree sum condition
    Opis:
    For an integer k ≥ 2, a k-tree T is defined as a tree with maximum degree at most k. If a k-tree T spans a graph G, then T is called a spanning k-tree of G. Since a spanning 2-tree is a Hamiltonian path, a spanning k-tree is an extended concept of a Hamiltonian path. The first result, implying the existence of k-trees in star-free graphs, was by Caro, Krasikov, and Roditty in 1985, and independently, Jackson and Wormald in 1990, who proved that for any integer k with k ≥ 3, every connected K1,k-free graph contains a spanning k-tree. In this paper, we focus on a sharp condition that guarantees the existence of a spanning k-tree in K1,k+1-free graphs. In particular, we show that every connected K1,k+1-free graph G has a spanning k-tree if the degree sum of any 3k−3 independent vertices in G is at least |G|−2, where |G| is the order of G.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 1; 5-13
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Nowhere-Zero Unoriented 6-Flows on Certain Triangular Graphs
    Autorzy:
    Yang, Fan
    Li, Liangchen
    Zhou, Sizhong
    Powiązania:
    https://bibliotekanauki.pl/articles/32309450.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    nowhere-zero k -flow
    triangle-tree
    triangle-star
    bidirected graph
    Opis:
    A nowhere-zero unoriented flow of graph G is an assignment of non-zero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A nowhere-zero unoriented k-flow is a flow with values from the set {±1, . . ., ±(k − 1)}, for short we call it NZ-unoriented k-flow. Let H1 and H2 be two graphs, H1⊕H2 denote the 2-sum of H1 and H2, if E(H1⊕H2) = E(H1) ∪ E(H2), |V(H1)∩V(H2)|=2, and |E(H1)∩E(H2)| = 1. A triangle-path in a graph G is a sequence of distinct triangles T1, T2, . . ., Tm in G such that for 1 ≤ i ≤ m, |E(Ti)∩E(Ti+1)| = 1 and E(Ti)∩E(Tj)=∅ if j>i+1. A triangle-star is a graph with triangles such that each triangle having one common edges with other triangles. Let G be a graph which can be partitioned into some triangle-paths or wheels H1, H2, . . ., Ht such that G = H1⊕H2⊕...⊕Ht. In this paper, we prove that G except a triangle-star admits an NZ-unoriented 6-flow. Moreover, if each Hi is a triangle-path, then G except a triangle-star admits an NZ-unoriented 5-flow.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 727-746
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-6 z 6

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