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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 88 of 349

  • Restricted completion of sparse partial Latin squares

  • Part of
  • Lina J. Andrén, Carl Johan Casselgren, Klas Markström
  • Journal:
    Combinatorics, Probability and Computing / Volume 28 / Issue 5 / September 2019
    Published online by Cambridge University Press:
    20 February 2019, pp. 675-695

  • An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

  • Improper colouring of graphs with no odd clique minor

  • Part of
  • Dong Yeap Kang, Sang-Il Oum
  • Journal:
    Combinatorics, Probability and Computing / Volume 28 / Issue 5 / September 2019
    Published online by Cambridge University Press:
    04 February 2019, pp. 740-754

  • As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

Page 88 of 349