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Register a new userDegree-regular triangulations of surfaces
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A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on $S$, and we list the possibilities. A key ingredient of the proof is to show that any two $d$-regular triangulations of the plane for $d> 6 $ are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combinatorial topology.
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We study $2k$-factors in $(2r+1)$-regular graphs. Hanson, Loten, and Toft proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a $2$-factor. We generalize their result by proving for $kle(2r+1)/3$ that every $(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor. Both the restriction on $k$ and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly $2r-3(k-1)+1$ cut-edges but no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have no $2k$-factor, as studied by Bollobas, Saito, and Wormald.
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
A two-dimensional simplicial complex is called $d$-{em regular} if every edge of it is contained in exactly $d$ distinct triangles. It is called $epsilon$-expanding if its up-down two-dimensional random walk has a normalized maximal eigenvalue which is at most $1-epsilon$. In this work, we present a class of bounded degree 2-dimensional expanders, which is the result of a small 2-complex action on a vertex set. The resulted complexes are fully transitive, meaning the automorphism group acts transitively on their faces. Such two-dimensional expanders are rare! Known constructions of such bounded degree two-dimensional expander families are obtained from deep algebraic reasonings (e.g. coset geometries). We show that given a small $d$-regular two-dimensional $epsilon$-expander, there exists an $epsilon=epsilon(epsilon)$ and a family of bounded degree two-dimensional simplicial complexes with a number of vertices goes to infinity, such that each complex in the family satisfies the following properties: * It is $4d$-regular. * The link of each vertex in the complex is the same regular graph (up to isomorphism). * It is $epsilon$ expanding. * It is transitive. The family of expanders that we get is explicit if the one-skeleton of the small complex is a complete multipartite graph, and it is random in the case of (almost) general $d$-regular complex. For the randomized construction, we use results on expanding generators in a product of simple Lie groups. This construction is inspired by ideas that occur in the zig-zag product for graphs. It can be seen as a loose two-dimensional analog of the replacement product.
We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $delta(G) ge frac{n+r-2}{2}$ when $nr equiv 0 pmod 2$. This answers a question of M.~Kriesell.
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