Connectivity and matching extendability of optimal $1$-embedded graphs on the torus

TL;DR

This work analyzes optimal $1$-embedded graphs on the torus ($O1TGs$), characterizing their connectivity in terms of the quadrangular subgraph $Q(G)$ and distinguishing the $8$-regular and non-$8$-regular cases. It proves that all $O1TGs are at least $4$-connected, with $8$-regular graphs yielding $ ext{$ ext{κ}(G)$}$ in $\{6,8 ight ight floor$ and an explicit criterion for $8$-connectedness via $Q(G)$ being non-isomorphic to $Q(p,r,3)$. For non-$8$-regular graphs, the paper identifies $ ext{$ ext{κ}(G)$}$ as $4$, $5$, or $6$ under precise $Q(G)$-structure conditions, including trivial and homotopic cycles. Building on these connectivity results, the authors derive exact characterizations of $1$-, $2$-, and $3$-extendability of $O1TGs$ in terms of $Q(G)$, showing that $3$-extendability occurs only for $8$-regular graphs. They also discuss potential extensions to the Klein bottle and propose conjectures guiding future study on other closed surfaces.

Abstract

In this paper, we discuss optimal $1$-toroidal graphs (abbreviated as O1TG), which are drawn on the torus so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n$ edges. We first consider connectivity of O1TGs, and give the characterization of O1TGs having connectivity exactly $k$ for each $k\in \{4, 5, 6, 8\}$. In our argument, we also show that there exists no O1TG having connectivity exactly $7$. Furthermore, using the result above, we discuss extendability of matchings, and give the characterization of $1$-, $2$- and $3$-extendable O1TGs in turn.

Figures (1)

• Figure 1: Homotopic closed walks $W_1, W_2$ and $W_2'$.