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  • 1.
    Andren, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed)
    Abstract [en]

    We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

  • 2.
    Andrén, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

  • 3.
    Asratian, Armen S.
    et al.
    Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On interval edge colorings of (a,b)-biregular bipartitie graphs2006In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 307, no 15, p. 1951-1956Article in journal (Refereed)
  • 4.
    Asratian, Armen S.
    et al.
    Linköping University Linköping Sweden.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On Path Factors of (3,4)-Biregular Bigraphs2008In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 24, no 5, p. 405-411Article in journal (Refereed)
    Abstract [en]

    A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

  • 5.
    Asratian, Armen S.
    et al.
    Linköping University, Linköping, Sweden.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vandenbussche, Jennifer
    Southern Polytechnic State University, Marietta, Georgia.
    West, Douglas B.
    University of Illinois, Urbana, Illinois.
    Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs2009In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, no 2, p. 88-97Article in journal (Refereed)
    Abstract [en]

    An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.

  • 6.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A note on path factors of (3,4)-biregular bipartite graphsManuscript (preprint) (Other academic)
  • 7.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A note on path factors of (3,4)-biregular bipartite graphs2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 18, no 1, p. P218-Article in journal (Refereed)
    Abstract [en]

    A proper edge coloring of a graph G with colors 1,2,3, ... is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graphis (3,4)-biregular if all vertices in one part have degree 3 and all vertices in the other part have degree 4. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph G has a spanning subgraph whose components are paths with end points at 3-valent vertices and lengths in {2,4,6,8}, then G has an interval coloring. It was also conjectured that every simple (3,4)-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple (3,4)-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at 3-valent vertices and lengths not exceeding 22.

  • 8.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Coloring graphs from random lists of fixed sizeManuscript (preprint) (Other academic)
  • 9.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Coloring graphs from random lists of size 2Manuscript (preprint) (Other academic)
  • 10.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Coloring graphs from random lists of size 22012In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 33, no 2, p. 168-181Article in journal (Refereed)
    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.

  • 11.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On avoiding some families of arrays2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 5, p. 963-972Article in journal (Refereed)
    Abstract [en]

    An n×n array A with entries from {1,…,n} is avoidable if there is an n×n Latin square L such that no cell in L contains a symbol that occurs in the corresponding cell in A. We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is NP-complete, even in the case when the array has entries from only two distinct symbols. Assuming that PNP, this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order n≥2k with entries from at most k distinct symbols and where each symbol occurs in at most n−2 cells is avoidable, and every single entry array of order n, where each of the symbols 1,…,n occurs in at most cells, is avoidable. Additionally, if k≥2, then every single entry array of order at least n≥4, where at most k rows contain non-empty cells and where each symbol occurs in at most nk+1 cells, is avoidable.

  • 12.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On some graph coloring problems2011Doctoral thesis, comprehensive summary (Other academic)
  • 13.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vertex coloring complete multipartite graphs from random lists of size 22011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 13, p. 1150-1157Article in journal (Refereed)
    Abstract [en]

    Let Ks×m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of Ks×m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring φ, such that φ(v)∈L(v) for all vV(Ks×m). We show that this property exhibits a sharp threshold at σ(m)=2(s−1)m.

  • 14. Casselgren, Carl Johan
    et al.
    Pham, Lan Anh
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Latin cubes of even order with forbidden entriesManuscript (preprint) (Other academic)
    Abstract [en]

    We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant γ>0 such that if n=2t and A is a 3-dimensional n×n×n array where every cell contains at most γn symbols, and every symbol occurs at most γn times in every line of A, then A is avoidable; that is, there is a Latin cube L of order n such that for every 1≤i,j,k≤n, the symbol in position (i,j,k) of L does not appear in the corresponding cell of A.

1 - 14 of 14
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