Realization of distance matrices by graphs of genus 1

来自有理矩阵的数值半群 IV：小 Frobenius 数或亏格的数值半群的矩阵维数计算

Numerical semigroups from rational matrices IV: computation of the matricial dimensions of numerical semigroups with small Frobenius number or genus

本文提出模论与线性规划方法，计算了所有 Frobenius 数≤10或亏格≤6的数值半群的矩阵维数，突破了以往技术局限。

亏格3的等面九边形八面体的整数实现

Integer Realization of an Equivelar Octahedron of Genus 3

本文给出了亏格3的八面九边形多面体曲面的整数坐标实现，具有24顶点36边，每对面共享边，验证了平面性、封闭可定向性及$C_4$对称性。

Polyhedra of genus 2 with 10 vertices and minimal coordinates

We give coordinate-minimal geometric realizations in general position of all 865 vertex-minimal triangulations of the orientable surface of genus 2 in the 4x4x4-cube.

Polyhedra of genus 3 with 10 vertices and minimal coordinates

We give coordinate-minimal geometric realizations in general position for 17 of the 20 vertex-minimal triangulations of the orientable surface of genus 3 in the 5x5x5-cube.

Surface realization with the intersection edge functional

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R^3 for all vertex-minimal triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12 vertices were obtained.

Z_2-genus of graphs and minimum rank of partial symmetric matrices

The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The \emph{$\mathbb{Z}_2$-genus} of a graph $G$, denoted by $\mathrm{g}_0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on $M_g$. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Štefankovič proved t…