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- 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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| 3. |
- Andrén, Daniel, 1973-
(författare)
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On the Ising problem and some matrix operations
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour. Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices). In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice. To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy. The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function. The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations. We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.
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| 4. |
- Hägglund, Jonas, 1982-
(författare)
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Snarks : Generation, coverings and colourings
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson's conjecture and Tutte's 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll's poem, a \emph{snark} is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis. The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order.In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph $G$ has a cycle of length at least $|V(G)|-9$ then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24.In Paper II we present a new algorithm for generating all non-isomorphic snarks with a given number of vertices. We generate all snarks on 36 vertices and less and study these with respect to various properties. We find that a number of conjectures on cycle covers and colourings holds for all graphs of these orders. Furthermore we present counterexamples to no less than eight published conjectures on cycle coverings, cycle decompositions and the general structure of regular graphs. In Paper III we show that Jaeger's Petersen colouring conjecture holds for three infinite families of snarks and that a minimum counterexample to this conjecture cannot contain a certain subdivision of $K_{3,3}$ as a subgraph. Furthermore, it is shown that one infinite family of snarks have strong Petersen colourings while another does not have any such colourings.Two simple constructions for snarks with arbitrary high oddness and resistance is given in Paper IV. It is observed that some snarks obtained from this construction have the property that they require at least five perfect matchings to cover the edges. This disproves a suggested strengthening of Fulkerson's conjecture.
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| 5. |
- Larsson, Joel, 1987-
(författare)
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On random satisfiability and optimization problems
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In Paper I, we study the following optimization problem: in the complete bipartite graph where edges are given i.i.d. weights of pseudo-dimension q>0, find a perfect matching with minimal total weight. The generalized Mézard-Parisi conjecture states that the limit of this minimum exists and is given by the solution to a certain functional equation. This conjecture has been confirmed for q=1 and for q>1. We prove it for the last remaining case 0<q<1.In Paper II, we study generalizations of the coupon collector problem. Versions of this problem shows up naturally in various context and has been studied since the 18th century. Our focus is on using existing methods in greater generality in a unified way, so that others can avoid ad-hoc solutions.Papers III & IV concerns the satisfiability of random Boolean formulas. The classic model is to pick a k-CNF with m clauses on n variables uniformly at random from all such formulas. As the ratio m/n increases, the formulas undergo a sharp transition from satisfiable (w.h.p.) to unsatisfiable (w.h.p.). The critical ratio for which this occurs is called the satisfiability threshold.We study two variations where the signs of variables in clauses are not chosen uniformly. In paper III, variables are biased towards occuring pure rather than negated. In paper IV, there are two types of clauses, with variables in them biased in opposite directions. We relate the thresholds of these models to the threshold of the classical model.
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| 6. |
- Pham, Lan Anh, 1991-
(författare)
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On avoiding and completing colorings
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- All of my papers are related to the problem of avoiding and completing an edge precoloring of a graph. In more detail, given a graph G and a partial proper edge precoloring φ of G and a list assignment L for every non-colored edge of G, can we extend φ to a proper edge coloring of G which avoids L? In Paper I, G is the d-dimensional hypercube graph Qd, a partial proper edge precoloring φ and a list assignment L must satisfy certain sparsity conditions. Paper II still deals with the hypercube graph Qd, but the list assignment L for every edge of Qd is an empty set and φ must be a partial proper edge precoloring of at most d-1 edges. In Paper III, G is a (d,s)-edge colorable graph; that is G has a proper d-edge coloring, where every edge is contained in at least s-1 2-colored 4-cycles, L must satisfy certain sparsity conditions and we do not have a partial proper edge precoloring φ on edges of G. The problem in Paper III is also considered in Paper IV and Paper V, but here G can be seen as the complete 3-uniform 3-partite hypergraph K3n,n,n, where n is a power of two in paper IV and n is an even number in paper V.
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