Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The point-thicknesses of complete graphs and complete multipartite graphs

Wenzhong Liu, Wangkai Zhang

TL;DR

The paper determines the exact point-thickness $\theta'(G)$ for two fundamental graph families: complete graphs and complete multipartite graphs. It develops a two-regime formula for complete multipartite graphs $G=K_{1^{k_1},2^{k_2},3^{k_3},p_1,\dots,p_n}$, depending on whether $p_0=k_1+2k_2+3k_3$ is at most $2n$ or greater, and uses the function $N(k_1,k_2)$ and parity considerations to express $\theta'(G)$ in closed form. The results yield the exact value $\theta'(K_n)=\lceil n/4 \rceil$ for complete graphs, and provide constructive upper-lower bound arguments via $I_2$-sets and non-planarity lemmas to establish tightness. These findings contribute precise partition-based characterizations of planarity-influenced graph decompositions, with potential implications for planarization and graph-drawing applications in combinatorial and computational contexts.

Abstract

The point-thickness $θ'(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ of $G$ is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thicknesses of complete graphs and complete multipartite graphs.

The point-thicknesses of complete graphs and complete multipartite graphs

TL;DR

The paper determines the exact point-thickness for two fundamental graph families: complete graphs and complete multipartite graphs. It develops a two-regime formula for complete multipartite graphs , depending on whether is at most or greater, and uses the function and parity considerations to express in closed form. The results yield the exact value for complete graphs, and provide constructive upper-lower bound arguments via -sets and non-planarity lemmas to establish tightness. These findings contribute precise partition-based characterizations of planarity-influenced graph decompositions, with potential implications for planarization and graph-drawing applications in combinatorial and computational contexts.

Abstract

The point-thickness of a graph is the minimum number of subsets into which the vertex set of is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thicknesses of complete graphs and complete multipartite graphs.
Paper Structure (2 sections, 5 theorems, 30 equations)

This paper contains 2 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1.1

Suppose that $k_1, k_2$, $k_3$ and $n$ are any nonnegative integers and that $p_0=k_1+2k_2+3k_3$. Let $G$ be a complete multipartite graph $K_{1^{k_{1}}, 2^{k_{2}}, 3^{k_{3}}, p_1, p_2, \cdots, p_n}$ with $4 \leq p_1\leq p_2\leq \cdots \leq p_n$. We have $(a)$ if $p_0 \leq 2n$, then $(b)$ if $p_0 > 2n$, then where $k_1+k_3\equiv \varepsilon$ (mod 2).

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3