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Note on the thickness of the Cartesian product of a complete graph and a path

Kenta Noguchi

Abstract

We determine the thickness of the Cartesian product $K_{6p+4} \square P_2$ for $p \ge 0$ and of the Cartesian product $K_8 \square P_m$ for $m \ge 1$, where $K_n$ and $P_m$ denote the complete graph on $n$ vertices and the path on $m$ vertices, respectively.

Note on the thickness of the Cartesian product of a complete graph and a path

Abstract

We determine the thickness of the Cartesian product for and of the Cartesian product for , where and denote the complete graph on vertices and the path on vertices, respectively.

Paper Structure

This paper contains 4 sections, 8 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proofs of the main theorems
  4. Concluding remarks

Key Result

Theorem 1

The thickness of $K_n$ is $\theta(K_n) = \left\lfloor \frac{n+7}{6} \right\rfloor$, except that $\theta(K_9)=\theta(K_{10})=3$.

Figures (2)

  • Figure 1: Planar graphs $H_1^m$ and $H_2^m$.
  • Figure 2: Planar graphs $I_1^m$ and $I_2^m$.

Theorems & Definitions (13)

  • Theorem 1: AG, BH
  • Theorem 2: Theorem 2 in CKY
  • Theorem 3: Theorem 3 in CKY
  • Theorem 4
  • Theorem 5
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['mainthm1']}
  • ...and 3 more