Partial-twuality polynomials of delta-matroids

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arXiv:2210.15273v1 [math.CO] 27 Oct 2022

Qi Yan

School of Mathematics

China University of Mining and Technology

P. R. China

Xian’an Jin1

School of Mathematical Sciences

Xiamen University

P. R. China

Email:qiyan@cumt.edu.cn; xajin@xmu.edu.cn

Abstract

Gross, Mansour and Tucker introduced the partial-twuality polynomial of a

ribbon graph. Chumutov and Vignes-Tourneret posed a problem: it would

be interesting to know whether the partial duality polynomial and the related

conjectures would make sense for general delta-matroids. In this paper we

consider analogues of partial-twuality polynomials for delta-matroids. Var-

ious possible properties of partial-twuality polynomials of set systems are

studied. We discuss the numerical implications of partial-twualities on a sin-

gle element and prove that the intersection graphs can determine the partial-

twuality polynomials of bouquets and normal binary delta-matroids, respec-

tively. Finally, we give a characterization of vf-safe delta-matroids whose

partial-twuality polynomials have only one term.

Keywords: Set system, delta-matroid, ribbon graph, twuality, polynomial

2020 MSC: 05B35, 05C10, 05C31

1. Introduction

In [16], Wilson found that the two long-standing duality operators δ(ge-

ometric duality) and τ(Petrie duality) generate a group of six ribbon graph

operators, that is, every other composition of δand τis equivalent to one of

1Corresponding author.

Preprint submitted to Elsevier

the ﬁve operators δ,τ,δτ,τδ,δτδ, or to the identity operator. Abrams and

Ellis-Monaghan [1] called the ﬁve operators twualities. The partial (geomet-

ric) dual with respect to a subset of edges of a ribbon graph was introduced

by Chmutov [7] in order to unify various connections between the Jones-

Kauﬀman and Bollob´as-Riordan polynomials. Ellis-Monaghan and Moﬀatt

[12] generalized this partial-duality construction to the other four operators,

which they called partial-twualities.

Gross, Mansour and Tucker [13,14] introduced the partial-twuality poly-

nomial for δ, τ, δτ, τδ, and δτδ. Various basic properties of partial-twuality

polynomials were studied, including interpolation and log-concavity. Re-

cently, Chumutov and Vignes-Tourneret [8] posed the following question:

Question 1. [8] Ribbon graphs may be considered from the point of view

of delta-matroid. In this way the concepts of partial (geometric) duality

and genus can be interpreted in terms of delta-matroids [9,10]. It would

be interesting to know whether the partial-δpolynomial and the related

conjectures would make sense for general delta-matroids.

In [18], we showed that the partial-δpolynomials have delta-matroid ana-

logues. We introduced the twist polynomials of delta-matroids and discussed

their basic properties for delta-matroids. Chun et al. [9] showed that the

loop complemenation is the delta-matroid analogue of partial Petriality. In

this paper we consider analogues of other partial-twuality polynomials for

delta-matroids.

This paper is organised as follows. In Section 2 we recall the deﬁnition

of partial-twuality polynomials of ribbon graphs. Analogously, we introduce

the partial-twuality polynomials of set systems. In Section 3, various pos-

sible properties of partial-twuality polynomials of set systems are studied.

In Section 4 we discuss the numerical implications of partial-twualities on a

single element and the interpolation. In Section 5, we prove that the inter-

section graphs can determine the partial-twuality polynomials of bouquets

and normal binary delta-matroids, respectively. Here we provide an answer

to the question [17]: can one derive something from bouquets that could

determine the partial-twuality polynomial completely. In Section 6 we give a

characterization of vf-safe delta-matroids whose partial-twuality polynomials

have only one term.

2. Preliminaries

2.1. Set systems and widths

Aset system is a pair D= (E, F) of a ﬁnite set Etogether with a

collection Fof subsets of E. The set Eis called the ground set and the

elements of Fare the feasible sets. We often use F(D) to denote the set of

feasible sets of D.Dis proper if F 6=∅, and is normal (respectively, dual

normal) if the empty set (respectively, the ground set) is feasible. The direct

sum of two set systems D= (E, F) and e

D= ( e

E, e

F) with disjoint ground

sets Eand e

E, written D⊕e

D, is deﬁned to be

D⊕e

D:= (E∪e

E, {F∪e

F:F∈ F and e

F∈e

F}).

As introduced by Bouchet in [3], a delta-matroid is a proper set system

D= (E, F) such that if X, Y ∈ F and u∈X∆Y, then there is v∈X∆Y

(possibly v=u) such that X∆{u, v} ∈ F. Here

X∆Y:= (X∪Y)−(X∩Y)

is the usual symmetric diﬀerence of sets. Note that the maximum gap in the

collection of sizes of feasible sets of a delta-matroid is two [15].

For a set system D= (E, F), let Fmax(D) and Fmin(D) be the collections

of maximum and minimum cardinality feasible sets of D, respectively. Let

Dmax := (E, Fmax(D)) and Dmin := (E, Fmin(D)). Let r(Dmax) and r(Dmin)

denote the sizes of largest and smallest feasible sets of D, respectively. The

width of D, denote by w(D), is deﬁned by

w(D) := r(Dmax)−r(Dmin).

For all non-negative integers i≤w(D), let

Fmax−i(D) = {F∈ F :|F|=r(Dmax)−i}

and

Fmin+i(D) = {F∈ F :|F|=r(Dmin) + i}.

2.2. Partial-twualities of set systems

We will consider the operations of twisting and loop complementation on

set systems. Twisting was introduced by Bouchet in [3], and loop comple-

mentation by Brijder and Hoogeboom in [5].

Let D= (E, F) be a set system. For A⊆E, the twist of Dwith respect

to A, denoted by D∗|A, is given by

(E, {A∆X:X∈ F}).

The ∗-dual of D, written D∗, is equal to D∗|E. Note that ∗-duality preserves

width. Throughout the paper, we will often omit the set brackets in the case

of a single element set. For example, we write D∗|einstead of D∗|{e}.

Let D= (E, F) be a set system and e∈E. Then D×|eis deﬁned to be

the set system (E, F′), where

F′=F∆{F∪e:F∈ F and e /∈F}.

If e1, e2∈Ethen

(D×|e1)×|e2= (D×|e2)×|e1.

This means that if A={e1,··· , em} ⊆ Ewe can unambiguously deﬁne the

loop complementation [5] of Don A, by

D×|A:= (···(D×|e1)×|e2···)×|em.

It is straightforward to show that the twist of a delta-matroid is a delta-

matroid [3], but the set of delta-matroids is not closed under loop com-

plementation (see, for example, [9]). Thus, we often restrict our attention

to a class of delta-matroids that is closed under loop complementation. A

delta-matroid D= (E, F) is said to be vf-safe [9] if the application of every

sequence of twists and loop complementations results in a delta-matroid.

In [5] it was shown that twists and loop complementations give rise to an

action of the symmetric group S3, with the presentation

S3∼

=B:=<∗,× | ∗2,×2,(∗×)3>,

on set systems. If D= (E, F) is a set system, e∈Eand a=a1a2···anis a

word in the alphabet {∗,×}, then

Da|e:= (···(Da1|e)a2|e···)an|e.

Note that the operators ∗and ×on diﬀerent elements commute [5]. If A=

{e1,··· , em} ⊆ E, we can unambiguously deﬁne

Da|A:= (···(Da|e1)a|e2···)a|em.

Let D1= (E, F) and D2be set systems. For • ∈ {∗,×,∗×,×∗,∗×∗}, we say

that D2is a partial-•dual of D1if there exists A⊆Esuch that D2=D1•|A.

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摘要：

arXiv:2210.15273v1[math.CO]27Oct2022Partial-twualitypolynomialsofdelta-matroidsQiYanSchoolofMathematicsChinaUniversityofMiningandTechnologyP.R.ChinaXian’anJin1SchoolofMathematicalSciencesXiamenUniversityP.R.ChinaEmail:qiyan@cumt.edu.cn;xajin@xmu.edu.cnAbstractGross,MansourandTuckerintroducedtheparti...

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