Pravděpodobnostní metody v diskrétní aplikované matematice

Fink, Jiří

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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