论文标题
在表面上的双图的程度序列上
On the degree sequences of dual graphs on surfaces
论文作者
论文摘要
给定两个图表$ g $和$ g^*$，边缘之间有一对一的对应关系，什么时候$ g $和$ g^*$形成一对双图，意识到嵌入在表面中的地图的顶点和国家？杰克·埃德蒙兹（Jack Edmonds）在1965年获得了标准。此外，让$ \ boldsymbol {d} =（d_1，\ ldots，d_n）$和$ \ boldsymbol {t} =（t_1，\ ldots，t_m）$是他们的度序列。然后，很明显，$ \ sum_ {i = 1}^n d_i = \ sum_ {j = 1}^m t_j = 2 \ ell $，其中$ \ ell $是两个图中每个图中的边缘数，而$χ= n- \ ell + m $是表面的特征。哪些序列$ \ boldsymbol {d} $和$ \ boldsymbol {t} $满足这些条件仍然无法实现为度序列？我们利用Edmonds的标准获得了球体的几个无限系列例外，$χ= 2 $和投射平面，$χ= 1 $。我们猜想$χ\ \ leq 0 $没有例外。
Given two graphs $G$ and $G^*$ with a one-to-one correspondence between their edges, when do $G$ and $G^*$ form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface? A criterion was obtained by Jack Edmonds in 1965. Furthermore, let $\boldsymbol{d}=(d_1,\ldots,d_n)$ and $\boldsymbol{t}=(t_1,\ldots,t_m)$ be their degree sequences. Then, clearly, $\sum_{i=1}^n d_i = \sum_{j=1}^m t_j = 2\ell$, where $\ell$ is the number of edges in each of the two graphs, and $χ= n - \ell + m$ is the Euler characteristic of the surface. Which sequences $\boldsymbol{d}$ and $\boldsymbol{t}$ satisfying these conditions still cannot be realized as the degree sequences? We make use of Edmonds' criterion to obtain several infinite series of exceptions for the sphere, $χ= 2$, and projective plane, $χ= 1$. We conjecture that there exist no exceptions for $χ\leq 0$.
