Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figure

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

Gibbs and Quantum Discrete Spaces

Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure