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Partitions of complete twisted graphs into plane spanning trees

Ana Paulina Figueroa, Eduardo Rivera-Campo

TL;DR

The authors characterize partitions of the complete twisted graph $T_{2n}$ into plane spanning trees, and show that partitions into isomorphic plane spanning trees consist precisely of balanced double stars, parameterized by a index $k$ via the family $A_k$ with exactly $n$ partitions. They establish a global counting framework: the number of partitions into arbitrary plane spanning trees satisfies $p_{n+1} = 5 \cdot 2^{2(n-2)} p_n$, yielding $p_n = 5^{n-2} 2^{1+(n-2)(n-3)}$ for $n \ge 2$. The results imply a complete structural description and enumeration of such partitions, and the paper derives a corollary that any complete topological graph on $n$ vertices contains a large subgraph admitting a partition into plane spanning trees, with $m \ge c \log^{1/8} n$. All mathematical notation is presented with proper $...$ delimiters.

Abstract

We characterize all partitions of the complete twisted graph $T_{2n}$ into plane spanning trees. In the case of partitions of $T_{2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double stars. As a consequence of our results, any complete topological graph with $n$ vertices contains a complete topological subgraph with $m \geq c\log^{1/8} n$ vertices that admits a partition into plane spanning trees.

Partitions of complete twisted graphs into plane spanning trees

TL;DR

The authors characterize partitions of the complete twisted graph into plane spanning trees, and show that partitions into isomorphic plane spanning trees consist precisely of balanced double stars, parameterized by a index via the family with exactly partitions. They establish a global counting framework: the number of partitions into arbitrary plane spanning trees satisfies , yielding for . The results imply a complete structural description and enumeration of such partitions, and the paper derives a corollary that any complete topological graph on vertices contains a large subgraph admitting a partition into plane spanning trees, with . All mathematical notation is presented with proper delimiters.

Abstract

We characterize all partitions of the complete twisted graph into plane spanning trees. In the case of partitions of into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double stars. As a consequence of our results, any complete topological graph with vertices contains a complete topological subgraph with vertices that admits a partition into plane spanning trees.
Paper Structure (4 sections, 8 theorems, 4 figures)

This paper contains 4 sections, 8 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary remarks
  3. Arbitrary partitions
  4. Partitions into isomorphic trees

Key Result

Lemma 3

Let $n \geq 2$ be an integer. If $\mathcal{S} = \{S_1, S_2, \ldots, S_{n}\}$ is a partition of the complete twisted graph $T_{2n}$ into plane spanning trees, then for $i=1, 2, \ldots, n-1$, the subgraph $S_i[V_i]$ of $S_i$ induced by the set of vertices $V_i = \{v_i, v_{i+1}, \ldots, v_n, w_n, w_{n-

Figures (4)

  • Figure 1: Complete twisted graph $T_6$.
  • Figure 2: A partition of $T_6$ into non-isomorphic plane trees (left) and a partition of $T_6$ into isomorphic plane trees (right).
  • Figure 3: Two partitions of $T_4$. $S_2=A_1$ on the left and $S_2=A_2$ on the right.
  • Figure 4: Partition of $T_6$ into balanced double stars $S_1,S_2, S_3$ with $S_3=A_2$. Here $v_3v_1 \in E(S_3)$ implies $v_3w_1 \in E(S_1)$ and $w_3v_2 \in E(S_3)$ implies $v_3v_2 \in E(S_2)$.

Theorems & Definitions (21)

  • Remark 1
  • proof
  • Remark 2
  • proof
  • Lemma 3
  • proof
  • Remark 4
  • proof
  • Theorem 5
  • proof
  • ...and 11 more