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Crossing number of graphs and $\mathsf{ΔY}$-move

Youngsik Huh, Ryo Nikkuni

Abstract

The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called $\mathsf{ΔY}$-move, on the complete graph $K_n$. Concretely it is shown that for any $k\in \mathbb{N}$, there exist a natural number $n$ and a sequence of $\mathsf{ΔY}$-moves $K_n\rightarrow G^{(1)}\rightarrow \cdots \rightarrow G^{(k)}$ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small $n$.

Crossing number of graphs and $\mathsf{ΔY}$-move

Abstract

The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called -move, on the complete graph . Concretely it is shown that for any , there exist a natural number and a sequence of -moves which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small .
Paper Structure (6 sections, 6 theorems, 33 equations, 14 figures)

This paper contains 6 sections, 6 theorems, 33 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Crossing-reducible trigons
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm3']}
  6. The intrinsic linkedness and crossing number of the Petersen family

Key Result

Theorem 1

For $n\geq 7$, a $\mathsf{\Delta Y}$-move on $K_n$ decreases the crossing number.

Figures (14)

  • Figure 1: $\mathsf{\Delta Y}$-move
  • Figure 2: Minimal-crossing drawings of the Petersen Family: The family consists of all graphs which are related to $K_6$ by $\mathsf{\Delta Y}$-moves.(The notations for the members follow HNTY.)
  • Figure 3: Minimal-crossing drawings of $K_7$, $G_7^{(1)}$, $G_7^{(2)}$, $G^*$ and Heawood graph
  • Figure 4:
  • Figure 5: $K_{2k+1}$ : Local pictures of $D$ around a vertex $w$.
  • ...and 9 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition
  • Lemma 4
  • Proposition 5
  • Lemma 6