Year: 2020
Type: preprint
Abstract: 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 $χ\leq 0$.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Topological and Algebraic Combinatorics, G287804571, G3559659542, G3652048868, The infrastructure group at University of Primorska, G4184417200, Topics in structural graph teory: invariants, separators and intersection representations, Algebra, discrete mathematics, probability and game theory, Traversability in vertex-transitive graphs
OpenAlex ID: W3046564315