Irreducibility of the Tutte polynomial of an embedded graph

Abstract

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.

The Tutte polynomial for graphs

“…In this section we introduce a Tutte polynomial for hypermaps. This polynomial is a direct generalisation of the well-studied Tutte polynomial for maps (which is also known as the ribbon graph polynomial) [4,12,19,22,25]. Although our polynomial is naturally defined in terms of deletion-contraction relations, it is more convenient to begin with an analogue of the dichromatic polynomial.…”

“…If H is a gem and therefore represents a graph embedded in surface then 𝑇 (H; 𝑥, 𝑦) coincides with the ribbon graph polynomial, also known as the 2-variable Bollobás-Riordan polynomial or the Tutte polynomial of cellularly embedded graphs. The ribbon graph polynomial is an important and wellstudied embedded graph analogue of the Tutte polynomial [12,19,22,25]. When H is a gem, 𝑇 (H; 𝑥, 𝑦) is also a specialisation of the Bollobás-Riordan polynomial of [4], with 𝑇 (H; 𝑥, 𝑦) = (𝑥 − 1) −𝛾 (H)/2 𝑅(H; 𝑥, 𝑦 − 1, √︁ (𝑥 − 1) (𝑦 − 1), 1).…”

“…To do this, since hypermaps generalize graphs cellularly embedded in surfaces, it is natural to build on the existing theory of embedded graph polynomials. There is a rich literature on analogues of the Tutte polynomial for embedded graphs (see, e.g., [4,12,19,22,25] and the survey [7]). By adapting the approach described in [25], we begin by constructing a version of the dichromatic polynomial for hypermaps.…”

“…Tutte polynomial for hypermaps, 𝑇 (H; 𝑥, 𝑦), should satisfy are: (1) it should be equivalent (up to change of variables and multiplication by simple prefactors) to the dichromatic polynomial; (2) it should satisfy the duality relation 𝑇 (H; 𝑥, 𝑦) = 𝑇 (H * ; 𝑦, 𝑥); and (3) 𝑇 (H; 𝑥, 𝑦) should coincide with the Tutte polynomial of an embedded graph[4,12,19,22,25] when H is a gem, and hence represents an embedded graph. For this we define…”