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  • 1. Aaghabali, M.
    et al.
    Akbari, S.
    Friedland, S.
    Markström, Klas
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Tajfirouz, Z.
    Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges2015Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 45, s. 132-144Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for even n. For odd n we state a conjecture on a sharp upper bound.

  • 2.
    Casselgren, Carl Johan
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Coloring graphs from random lists of size 22012Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 33, nr 2, s. 168-181Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let G = G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Delta. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set e of size sigma (n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring phi such that phi(v) is an element of L(v) for all v is an element of V(G). In particular, we show that if g is odd and sigma (n) = omega(n(1/(2g-2))), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n --> infinity. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n >= g, there is a graph H = H(n, g) with bounded maximum degree and girth g, such that if sigma (n) = 0(n(1/(2g-2))), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n --> infinity. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size sigma (n), exhibits a sharp threshold at sigma (n) = 2n. (C) 2011 Elsevier Ltd. All rights reserved.

  • 3. Christofides, Demetres
    et al.
    Markström, Klas
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The range of thresholds for diameter 2 in random Cayley graphs2014Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 35, s. 141-154Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Given a group G, the model g(G, p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. Given a family of groups (G(k)) and a c is an element of R+ we say that c is the threshold for diameter 2 for (G(k)) if for any epsilon > 0 with high probability Gamma is an element of g(G(k), p) has diameter greater than 2 if p <= root(c - epsilon)log n/n and diameter at most 2 if p >= root(c + epsilon)log n/n. In Christofides and Markstrom (in press) [5] we proved that if c is a threshold for diameter 2 for a family of groups (G(k)) then c is an element of [1/4, 2] and provided two families of groups with thresholds 1/4 and 2 respectively. In this paper we study the question of whether every c is an element of [1/4, 2] is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every c is an element of [1/4, 4/3] is a threshold but a c is an element of (4/3, 2] is a threshold if and only if it is of the form 4n/(3n - 1) for some positive integer n. 

  • 4.
    Häggkvist, Roland
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Denley, Tristan
    Completing some partial Latin squares2000Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 21, nr 7, s. 877-880Artikel i tidskrift (Refereegranskat)
  • 5.
    Markström, Klas
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Rucinski, Andrzej
    Perfect matchings (and Hamilton cycles) in hypergraphs with large degrees2011Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 32, nr 5, s. 677-687Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We establish a new lower bound on the l-wise collective minimum degree which guarantees the existence of a perfect matching in a k-uniform hypergraph, where 1 <= l < k/2. For l = 1, this improves a long-standing bound of Daykin and Haggkvist (1981) [5]. Our proof is a modification of the approach of Han et al. (2009) from [12]. In addition, we fill a gap left by the results solving a similar question for the existence of Hamilton cycles. (C) 2011 Published by Elsevier Ltd

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