A Tutte Polynomial for Maps

Let be a multigraph with for each vertex a cyclic order of the edges incident with it. For 3 , let be the dihedral group of order 2 . Define . Goodall et al in 2016 asked whether admits a nowhere‐identity ‐flow if and only if it admits a nowhere‐identity ‐flow with (a “nowhere‐identity dihedral ‐flow”). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere‐identity 2 ‐flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.

“…In fact, the following theorem of DeVos (but taken as it is stated in ) shows that if the commutator subgroup of

G

is large enough, then a plane‐sided bridge is the only obstruction. In the theorem,

Q_{8}

denotes the quaternion group.…”

Section: Negative Resultsmentioning

confidence: 99%

“…It was also mentioned in the introduction that both the number of nowhere‐zero

n

‐flows and the number of nowhere‐zero

G

‐flows are counted by a polynomial in

n

. Example

5.11

of demonstrates that the number of nowhere‐identity

D_{2 n}

‐flows in an embedded graph is a quasi‐polynomial in

n

of period

2

. The authors of asked if the number of nowhere‐identity dihedral

n

‐flows is also counted by a quasi‐polynomial in

n

of period

2

.…”

Section: Negative Resultsmentioning

confidence: 99%

“…It is not immediately clear how a counterpart of Theorem would look like for nonabelian groups. In a question is posed for the dihedral group

D_{2 n}

, with

n \geq 3

. By

Z_{n}

we denote the cyclic group of order

n

.…”

Section: Introductionmentioning

confidence: 99%

“…A nowhere‐identity

double-struckD

‐flow on

normalΓ

(defined formally in Section ) which takes values in

D^{< n}

, is called a nowhere‐identity dihedral

n

‐ flow . Goodall et al asked whether the following statement is true:

Γ normala normald normalm normali normalt normals normala normaln normalo normalw normalh normale normalr normale normal‐ normali normald normale normaln normalt normali normalt normaly D_{2 n} ‐ normalf normall normalo normalw \Leftrightarrow Γ normala normald normalm normali normalt normals normala normaln normalo normalw normalh normale normalr normale normal‐ normali normald normale normaln normalt normali normalt normaly normald normali normalh normale normald normalr normala normall n ‐ normalf normall normalo normalw .

The “

\Leftarrow

” direction is easily seen to hold. Indeed, observe that by composition with

Π_{n}

, a nowhere‐identity dihedral

n

‐flow yields a nowhere‐identity

D_{2 n}

‐flow.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

On dihedral flows in embedded graphs

Litjens

2018

Partition functions and a generalized coloring‐flow duality for embedded graphs

Litjens

Sevenster

2017

Let G be a finite group and χ:G→C a class function. Let H=(V,E) be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection F of faces of H. Define the partition function 0truePχ(H):=∑κ:E→G∏v∈Vχ(κfalse(δ(v)false)), where κ(δ(v)) denotes the product of the κ‐values of the edges incident with v (in cyclic order), where the inverse is taken for edges leaving v. Write 0trueχ=∑λmλχλ, where the sum runs over irreducible representations λ of G with character χλ and with mλ∈C for every λ. When H is connected, it is proved that 0truePχ(H)=false|Gfalse|false|Efalse|∑λχλfalse(1false)false|Ffalse|−false|Efalse|mλfalse|Vfalse|, where 1 is the identity element of G. Among the corollaries, a formula for the number of nowhere‐identity G‐flows on H is derived, generalizing a result of Tutte. We show that these flows correspond bijectively to certain proper G‐colorings of a covering graph of the dual graph of H. This correspondence generalizes coloring‐flow duality for planar graphs.

Nonabelian flows in networks

Gent

2023

In this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common constraint is conservation in a vertex, meaning that the sum of the labels on the incoming edges of this vertex equals the sum of those on the outgoing edges. One easy fact is that if a flow is conserving in all but one vertex, then it is also conserving in the remaining one. In our generalization we do not label the edges with real numbers, but with elements from an arbitrary group, where this fact becomes false in general. As we will show, graphs with the property that conservation of a flow in all but one vertex implies conservation in all vertices are precisely the planar graphs.