| 1. |
- Asratian, Armen, et al.
(författare)
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Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs
- 2009
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Ingår i: Journal of Graph Theory. - : Wiley Periodicals Inc.. - 0364-9024 .- 1097-0118. ; 61:2, s. 88-97
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Andrén, Lina J., 1980-
(författare)
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Avoidability by Latin squares of arrays of even order
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove that for any k and any 2k × 2k array A such that no cell in A contains more than k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.
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| 3. |
- Andrén, Lina J., 1980-
(författare)
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Avoidability of random arrays
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.
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| 4. |
- Andrén, Lina J., 1980-, et al.
(författare)
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Avoiding arrays of odd order by Latin squares
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 5. |
- Andrén, Lina J., 1980-
(författare)
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Avoiding (m, m, m)-arrays of order n = 2k
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.
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| 6. |
- Andrén, Lina J., 1980-
(författare)
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On Latin squares and avoidable arrays
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of the four papers listed below and a survey of the research area. I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k II Lina J. Andrén: Avoidability of random arrays III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2k . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.
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| 7. |
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| 8. |
- Fleischner, Herbert, et al.
(författare)
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Circuit double covers in special types of cubic graphs
- 2009
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Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 309:18, s. 5724-5728
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Tidskriftsartikel (refereegranskat)abstract
- Suppose that a 2-connected cubic graph G of order n has a circuit C of length at least n−4 such that G−V(C) is connected. We show that G has a circuit double cover containing a prescribed set of circuits which satisfy certain conditions. It follows that hypohamiltonian cubic graphs (i.e., non-hamiltonian cubic graphs G such that G−v is hamiltonian for every v∈V(G)) have strong circuit double covers.
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| 9. |
- Leader, Imre, et al.
(författare)
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Uncountable families of vertex-transitive graphs of finite degree
- 2006
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Ingår i: Discrete Mathematics. ; 306:7, s. 678-679
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Tidskriftsartikel (refereegranskat)abstract
- Recently the following question was relayed to the second author: What is the cardinality of the set of vertex transitive graphs of finite degree? Our aim in this short note is to show that there are $2^{\aleph_{0}}$ such graphs.
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| 10. |
- Öhman, Lars-Daniel
(författare)
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Partial latin squares are avoidable
- 2011
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Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 15:3, s. 485-497
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Tidskriftsartikel (refereegranskat)abstract
- A square array is avoidable if for each set of n symbols there is an n x n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m >= 2 any partial Latin square of order 4m - 1 is avoidable, thus concluding the proof that any partial Latin square of order at least 4 is avoidable.
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