Toroidal Cartesian Products Where One Factor is 3-Connected

TL;DR

We study the toroidal embeddings of Cartesian products $G \boxempty H$ where the factor $G$ is $3$-connected. Our approach combines Euler characteristic bounds and minor arguments to restrict the second factor to $P_2$ and derive that $G \boxempty H$ is toroidal only when $H = P_2$ and $G$ is outer-cylindrical, with a constructive embedding provided in the other direction. A key by-product is that $\gamma(K_4 \boxempty P_3) = 2$, and we exhibit infinite families such as $C^2_{2n} \boxempty P_2$ that realize toroidal products. These results illuminate how vertex connectivity constrains toroidal embeddability of Cartesian products and contribute to broader classification efforts for toroidal Cartesian products.

Abstract

In this paper, we show that if $G$ is $3$-connected, then the Cartesian product of graphs $G \square H$ embeds on the torus if and only if $G$ is outer-cylindrical and $H$ is a path on two vertices, $P_2$. As a by-product of our work, we also show that $K_{4} \square P_{3}$ has genus two.

Theorems & Definitions (19)

• theorem 1
• proposition thmcounterproposition
• proposition thmcounterproposition
• proof
• corollary thmcountercorollary
• lemma thmcounterlemma
• proof
• proposition thmcounterproposition
• proof
• lemma thmcounterlemma