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On the constant partial-dual polynomials of hypermaps

Yibing Xiang, Qi Yan

TL;DR

This work extends partial duality to hypermaps by introducing the partial-dual polynomial $^{\partial}\varepsilon_{H}(z)$ and establishes its fundamental properties through arrow presentations and genus calculations. It provides a precise characterization for prime connected ribbon hypermaps: $^{\partial}\varepsilon_{H}(z)$ is constant if and only if $H$ is a plane ribbon hypermap with a single hyperedge, linking constant-term behavior to planarity. The paper also analyzes special cases such as plane hyper-bouquets and hypertrees, giving exact forms like $^{\partial}\varepsilon_{B}(z)=2^{e(B)}$ and $^{\partial}\varepsilon_{H}(z)=2^{e(H)}$, and discusses the behavior of the polynomial under hypermap operations. Together, these results establish a combinatorial framework connecting partial duality, Euler-genus, and the structure of hypermaps through arrow representations.

Abstract

In this paper, we introduce the partial-dual polynomial for hypermaps, extending the concept from ribbon graphs. We discuss the basic properties of this polynomial and characterize it for hypermaps with exactly one hypervertex containing a non-zero constant term. Additionally, we show that the partial-dual polynomial of a prime connected hypermap $H$ is constant if and only if $H$ is a plane hypermap with a single hyperedge.

On the constant partial-dual polynomials of hypermaps

TL;DR

This work extends partial duality to hypermaps by introducing the partial-dual polynomial and establishes its fundamental properties through arrow presentations and genus calculations. It provides a precise characterization for prime connected ribbon hypermaps: is constant if and only if is a plane ribbon hypermap with a single hyperedge, linking constant-term behavior to planarity. The paper also analyzes special cases such as plane hyper-bouquets and hypertrees, giving exact forms like and , and discusses the behavior of the polynomial under hypermap operations. Together, these results establish a combinatorial framework connecting partial duality, Euler-genus, and the structure of hypermaps through arrow representations.

Abstract

In this paper, we introduce the partial-dual polynomial for hypermaps, extending the concept from ribbon graphs. We discuss the basic properties of this polynomial and characterize it for hypermaps with exactly one hypervertex containing a non-zero constant term. Additionally, we show that the partial-dual polynomial of a prime connected hypermap is constant if and only if is a plane hypermap with a single hyperedge.
Paper Structure (4 sections, 11 theorems, 32 equations, 7 figures)

This paper contains 4 sections, 11 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial duality for ribbon hypermaps
  4. The partial-dual polynomial

Key Result

Theorem 10

Let $H$ be a ribbon hypermap and $A\subseteq E(H)$. Then

Figures (7)

  • Figure 1: Construction \ref{['cons01']}
  • Figure 2: The four common line segments $cl_1, cl_2, cl_3$, and $cl_4$ alternate
  • Figure 3: Remark \ref{['remark1']}
  • Figure 4: Remark \ref{['remark2']}
  • Figure 5: SR(H)
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 2: BSM
  • Definition 3: BSM
  • Remark 5
  • Definition 6
  • Example 7
  • Definition 8: BSM
  • Remark 9
  • Theorem 10
  • proof
  • Corollary 11
  • ...and 24 more