[Paper Review] Generalized Petersen Graphs and Kronecker Covers

Matjaž Krnc, Tomaž Pisanski|arXiv (Cornell University)|Sep 26, 2019

Finite Group Theory Research5 citations

TL;DR

This paper characterizes which generalized Petersen graphs G(n,k) are Kronecker covers—bipartite double covers formed via the tensor product with K₂—by identifying exact conditions on parameters n and k. It proves that G(10,3) is the only such graph that is a Kronecker cover of two non-isomorphic graphs (the Petersen graph and H), while all others are either Kronecker covers of a single quotient graph or not Kronecker covers at all, with the quotient structure fully described using LCF notation.

ABSTRACT

Abstract: The family of generalised Petersen graphs G (n, k), introduced by Coxeter et al. [4] and named by 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 KC (G) of a simple undirected graph G is a special type of bipartite covering graph of G, isomorphic to the direct (tensor) product of G and K2. We characterize all generalised Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalised Petersen graphs, and describe all such pairs. The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.

Motivation & Objective

• To determine which generalized Petersen graphs G(n,k) are Kronecker covers of some graph.
• To characterize the structure of the quotient graphs obtained when G(n,k) is a Kronecker cover.
• To identify whether a generalized Petersen graph can be a Kronecker cover of more than one non-isomorphic graph.
• To describe the quotient graphs in terms of LCF notation and classify them based on the parity and congruence conditions of n and k.
• To establish the uniqueness of the quotient graph for each Kronecker cover, except for the exceptional case of G(10,3).

Proposed method

• Use the Kronecker cover construction as the tensor product G × K₂, defining a bipartite double cover with twice the number of vertices.
• Apply the characterization from Imrich and Pisanski (2007) that a bipartite graph is a Kronecker cover if and only if its automorphism group admits a fixed-point-free, color-reversing involution (Kronecker involution).
• Analyze the automorphism structure of G(n,k) to determine when such an involution exists, focusing on cases where n ≡ 0 or 2 mod 4 and k is odd.
• Define and use LCF notation for cubic Hamiltonian graphs to represent the quotient graphs, specifically introducing C⁺(n,k) and C⁻(n,k) as families of graphs derived from the involution structure.
• Prove uniqueness of the quotient graph by analyzing equivalence classes of Kronecker involutions using additive group structures and GCD-based orbit counting.
• Use case analysis based on k mod 4 (k ≡ 1 or 3 mod 4) to distinguish between the C⁺(n,k) and C⁻(n,k) quotient families under the condition k² ≡ 1 mod n and n divides (k²−1)/2.

Experimental results

Research questions

• RQ1 For which parameters (n,k) is the generalized Petersen graph G(n,k) a Kronecker cover of some graph?
• RQ2 Can a generalized Petersen graph be a Kronecker cover of more than one non-isomorphic graph?
• RQ3 What is the structure of the quotient graph when G(n,k) is a Kronecker cover?
• RQ4 Under what conditions on n and k is the quotient graph itself a generalized Petersen graph?
• RQ5 Are the quotient graphs unique for each Kronecker cover, or can multiple non-equivalent involutions yield isomorphic quotients?

Key findings

• G(10,3) is the only generalized Petersen graph that is a Kronecker cover of two non-isomorphic graphs: the Petersen graph and the graph H.
• When n ≡ 2 (mod 4) and k is odd, G(n,k) is a Kronecker cover; the quotient is G(n/2, k) if 4k < n, and G(n/2, n/2 − k) if n < 4k < 2n.
• When n ≡ 0 (mod 4) and k is odd, G(n,k) is a Kronecker cover if and only if n divides (k²−1)/2 or (n,k) = (8,3); the quotient is C⁺(n,k) if k ≡ 1 (mod 4), and C⁻(n,k) if k ≡ 3 (mod 4).
• For all other cases (e.g., n even and k even, or n odd), G(n,k) is not a Kronecker cover.
• The quotient graph is always unique for each Kronecker cover, except for G(10,3), which has two distinct quotient graphs.
• KC(G(n,k)) is itself a generalized Petersen graph if and only if n is odd, which implies that for even n, the Kronecker cover is not a generalized Petersen graph.