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Hamiltonicity of hypercubes without k-snakes and k-coils
Pěgřímek, David ; Gregor, Petr (advisor) ; Fink, Jiří (referee)
A snake (coil) is an induced path (cycle) in a hypercube. They are well known from the snake-in-the-box (coil-in-the-box) problem which asks for the longest snake (coil) in a hypercube. They have been generalized to k-snakes (k-coils) which preserve distances between their every two vertices at distance at most k − 1 in hypercube. We study them as a variant of Locke's hypothesis. It states that a balanced set F ⊆ V (Qn) of cardinality 2m can be avoided by a Hamiltonian cycle if n ≥ m + 2 and m ≥ 1. We show that if S is a k-snake (k-coil) in Qn for n ≥ k ≥ 6 (n ≥ k ≥ 7), then Qn − V (S) is Hamiltonian laceable. For a fixed k the number of vertices of a k-coil may even be exponential with n. We introduce a dragon, which is an induced tree in a hypercube, and its generalization a k-dragon which preserves distances between its every two vertices at distance at most k−1 in hypercube. By proving a specific lemma from my Bachelor thesis that was previously verified by a computer, we finish the proof of the theorem regarding Hamiltonian laceability of hypercubes without n-dragons.
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Probabilistic Methods in Discrete Applied Mathematics
Fink, Jiří ; Loebl, Martin (advisor) ; Koubek, Václav (referee) ; Sereni, Jean-Sébastein (referee)
One of the basic streams of modern statistical physics is an effort to understand the frustration and chaos. The basic model to study these phenomena is the finite dimensional Edwards-Anderson Ising model. We present a generalization of this model. We study set systems which are closed under symmetric differences. We show that the important question whether a groundstate in Ising model is unique can be studied in these set systems. Kreweras' conjecture asserts that any perfect matching of the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle. We prove this conjecture. The {\it matching graph} $\mg{G}$ of a graph $G$ has a vertex set of all perfect matchings of $G$, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph $\mg{Q_n}$ is bipartite and connected for $n \ge 4$. This proves Kreweras' conjecture that the graph $M_n$ is connected, where $M_n$ is obtained from $\mg{Q_n}$ by contracting all vertices of $\mg{Q_n}$ which correspond to isomorphic perfect matchings. A fault-free path in $Q_n$ with $f$ faulty vertices is said to be \emph{long} if it has length at least $2^n-2f-2$. Similarly, a fault-free cycle in $Q_n$ is long if it has length at least $2^n-2f$. If all faulty vertices are...
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Optimization and Statistics
Fink, Jiří ; Loebl, Martin (advisor)
CONTENTS Nazev prace: Autor: Katedra (ustav-): Vedouci diplomove prace: E-mail vedouciho: Kh'cova slova; Abstrakt: Optimization and Statistics Jifi Fink Katedra aplikovane matematiky Doc. RNDr. Martin Loebl, CSc. loebl@kam.mff, cuni.cz Edwards-Anderson Ising model, Teorie grafu, T-join, Gaussovska distribuce Jedni'm ze zakladnich problemu moderni statisticke fyzikj' je'snada porozumet frus- traci a chaosu. Zakladnfm modelem je konecne dimenzionalni Edwards-Anderson Ising model. V optimalizaci to odpovida zkoumam minimalni'ch T-joinu v konecnych mnzkach s nahodnymi vahami na hranach. V teto praci studujeme "random join", coz je nahodna cesta mezi dvema pevne danj^mi \Tcholy. Puvodni definice je pfilis slozita; a tak jsme ukazali jednodussi. Tato defmice je pouzita k pfesnernu vypoctu "random join" na kruznici. Take jsme ukazali specialm algoritmus, ktery hleda cestu v mrfzce s danymi hranami. Tento algoritmus muze byt pouzit k experimentalnimu studovani "random join". Title: Author: Department: Supervisor: Supervisor's e-mail address: Keywords: Abstract: Optimization and Statistics Jin Fink Department of Applied Mathematics Doc. RNDr. Martin Loebl, CSc. loebl@kam.mff.cuni.cz Edwards-Anderson Ising model, Graph theory, T-join, the Gaussian distribution One of the basic streams of modern statistics physics is...
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Optimization and Statistics
Fink, Jiří ; Kratochvíl, Jan (referee) ; Loebl, Martin (advisor)
One of the basic streams of modern statistics physics is an effort to understand the frustration and chaos. The basic model to study these phenomena is the finite dimensional Edwards-Anderson Ising model. In discrete optimisation this corresponds to the minimal T-joins in a finite lattice with random weights of edges. This thesis studies a random join which is a random path between two given vertices. The original definition of the random join is very complex, and we have managed to find an equivalent one which is more natural. We use our definition to exactly compute the random join on circles. We also propose an algorithm which finds the shortest path in a large lattice with given weights of edges. This algorithm can be used for an experimental study of the random join.
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