[Paper Review] Characterization of generalized Petersen graphs that are Kronecker covers

Matjaž Krnc, Tomaž Pisanski|arXiv (Cornell University)|Feb 20, 2018

Finite Group Theory Research4 citations

TL;DR

This paper characterizes all generalized Petersen graphs $G(n,k)$ that are Kronecker covers—bipartite double covers formed via the tensor product with $K_2$—and determines the structure of their quotient graphs. The key result is a complete classification: $G(n,k)$ is a Kronecker cover if and only if $k^2 \equiv 1 \pmod{n}$ and $n \mid \frac{k^2 - 1}{2}$, with the quotient graph being isomorphic to a specific LCF-notation graph $C^+(n,k)$ or $C^-(n,k)$ depending on the parity of $k$. The classification reveals that such graphs are Kronecker covers precisely when $n$ is odd, and the quotient is unique under these conditions.

ABSTRACT

The family of generalized Petersen graphs $G(n, k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $\mathrm{KC}(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.

Motivation & Objective

• To determine all generalized Petersen graphs $G(n,k)$ that are Kronecker covers of some graph.
• To characterize the structure of the quotient graphs obtained from the Kronecker cover operation on $G(n,k)$.
• To classify the parameters $(n,k)$ for which $G(n,k)$ is a Kronecker cover, particularly focusing on the conditions under which the quotient is unique.
• To explore the relationship between generalized Petersen graphs and their Kronecker covers, especially when the cover is itself a generalized Petersen graph.

Proposed method

• The paper uses the Kronecker cover construction, defined as the tensor product $G \times K_2$, to generate bipartite double covers of generalized Petersen graphs.
• It analyzes Kronecker involutions—automorphisms of order two that define the quotient graphs under the cover operation.
• The authors define two families of graphs, $C^+(n,k)$ and $C^-(n,k)$, using LCF notation to represent the quotient graphs of $G(n,k)$ under the Kronecker cover.
• The classification relies on solving the congruence conditions $k^2 \equiv 1 \pmod{n}$ and $n \mid \frac{k^2 - 1}{2}$, which determine when $G(n,k)$ admits a Kronecker cover.
• Group-theoretic arguments are used to show that the set of Kronecker involutions partitions into equivalence classes, and the number of such classes is shown to be one under the stated conditions.
• The proof uses additive group structures over $\mathbb{Z}/m\mathbb{Z}$ to analyze the orbits of involutions and establish uniqueness of the quotient graph.

Experimental results

Research questions

• RQ1 For which parameters $(n,k)$ is the generalized Petersen graph $G(n,k)$ a Kronecker cover of another graph?
• RQ2 What is the structure of the quotient graph when $G(n,k)$ is a Kronecker cover?
• RQ3 Under what conditions is the quotient graph of $G(n,k)$ under the Kronecker cover unique?
• RQ4 When is the Kronecker cover $\mathrm{KC}(G(n,k))$ itself a generalized Petersen graph?
• RQ5 How do the symmetries and automorphisms of $G(n,k)$ relate to the Kronecker cover and its quotient?

Key findings

• A generalized Petersen graph $G(n,k)$ is a Kronecker cover if and only if $k^2 \equiv 1 \pmod{n}$ and $n \mid \frac{k^2 - 1}{2}$.
• When $k \equiv 1 \pmod{4}$, the quotient graph of $G(n,k)$ under the Kronecker cover is isomorphic to $C^+(n,k)$, defined via LCF notation.
• When $k \equiv 3 \pmod{4}$, the quotient graph is isomorphic to $C^-(n,k)$, also defined via LCF notation.
• The quotient graph is unique under the given conditions, as the number of equivalence classes of Kronecker involutions is exactly one.
• The Kronecker cover $\mathrm{KC}(G(n,k))$ is itself a generalized Petersen graph if and only if $n$ is odd.
• For even $n$, the Kronecker cover $\mathrm{KC}(G(n,k))$ is not a generalized Petersen graph, and falls into two known classes depending on the parity of $k$.