[Paper Review] Graphs as rotations

Dainis Zeps|ArXiv.org|Sep 1, 2009

Graph Labeling and Dimension Problems9 references3 citations

TL;DR

This paper introduces a novel algebraic framework for representing graphs as permutations, where combinatorial maps are defined as pairs of permutations encoding vertex and face cycles. The key contribution is showing that every combinatorial map can be uniquely expressed as the product of its knot (a permutation encoding zigzag walks) and a self-conjugate map, establishing a group-theoretic foundation for graph embeddings on surfaces.

ABSTRACT

Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces. Further, we define multiplication of these objects, that coincides with the multiplication of permutations. We consider closed under multiplication classes of combinatorial maps that consist of closed classes of combinatorial maps with fixed edges where each such class is defined by a knot. One class among them is special, containing selfconjugate maps.

Motivation & Objective

• To establish a formal algebraic representation of graphs with fixed rotations using permutation pairs.
• To define a multiplication operation on combinatorial maps that preserves closure under the operation.
• To introduce the concept of a combinatorial knot as a cycle structure derived from alternating edge and corner traversals.
• To characterize self-conjugate maps as a subgroup within the class of maps with fixed edges.
• To prove that every map can be factored into its knot and a self-conjugate map, enabling structural decomposition.

Proposed method

• Define combinatorial maps as pairs of differing permutations (P, Q) such that P⁻¹Q is a matching (i.e., a fixed-point-free involution).
• Introduce the next-edge-matching π = P⁻¹Q and the edge-matching ϱ = πᴾ⁻¹ to encode edge and corner relationships.
• Define the knot μ as a permutation that alternates between π and ϱ actions on corners, forming cycles of even length.
• Establish multiplication of maps via permutation multiplication, showing closure under this operation when π is fixed.
• Use conjugation and coset decomposition to analyze isomorphism classes and show that isomorphic maps are conjugate under elements of the stabilizer subgroup Kπ.
• Prove that the class of self-conjugate maps forms a subgroup and that every map factors uniquely as μ × s, where μ is the knot and s is self-conjugate.

Experimental results

Research questions

• RQ1 How can graphs with fixed rotation be algebraically represented using permutation pairs?
• RQ2 What algebraic structure underlies the set of combinatorial maps closed under multiplication?
• RQ3 How is the combinatorial knot related to zigzag walks and edge-vertex traversal patterns?
• RQ4 Can every combinatorial map be decomposed into a knot and a self-conjugate map?
• RQ5 Under what conditions are two maps isomorphic, and how does this relate to conjugation in the permutation group?

Key findings

• Every combinatorial map (P, P·π) admits a well-colored partition of corners into two sets induced by the next-edge-matching π and edge-matching ϱ.
• The knot μ, defined by alternating actions of π and ϱ, is a permutation whose cycles correspond to zigzag walks and is invariant under direction reversal.
• The knot μ satisfies the identity π^μ = ϱ, establishing a direct algebraic link between the edge and next-edge matchings.
• The class of maps with fixed π forms a group under multiplication, isomorphic to the symmetric group S₂ₘ, and decomposes into (2m−1)!! left cosets of the stabilizer subgroup Kπ.
• Every map can be uniquely expressed as the product of its knot μ and a self-conjugate map s, i.e., P = μ × s, proving a structural factorization theorem.
• Isomorphic maps are conjugate via a permutation A such that A ∈ Kπ, and the maps belong to the same closed multiplication class if and only if A stabilizes π.