The thickness of a simple graph G is the smallest number of...

The thickness of a simple graph $G$ is the smallest number of planar subgraphs of $G$ that have $G$ as their union. $$ \begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { and no circuits of length }} \\ {\text { three, then the thickness of } G \text { is at least }[e /(2 v-4)]}\end{array} $$

The thickness of a simple graph $G$ is the smallest number of planar subgraphs of $G$ that have $G$ as their union.
$$
\begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { and no circuits of length }} \\ {\text { three, then the thickness of } G \text { is at least }[e /(2 v-4)]}\end{array}
$$

Key Concepts

Graph Thickness

Graph thickness is the minimum number of planar subgraphs into which a given graph can be decomposed such that their union recovers the original graph. This concept is important in understanding how non-planar graphs can be studied through their planar components and has implications in areas like network layout problems, where reducing edge crossings is crucial.

Cycle (Circuit) in Graphs

In graph theory, a circuit or cycle is a closed path where no edges (and in simple cycles no vertices) are repeated, except for the starting and ending vertex being the same. The characterization of cycles, particularly restrictions on their lengths such as excluding triangles, influences the properties and structure of a graph, affecting both its local and global properties.

Simple Graph

A simple graph is a type of graph that does not allow multiple edges between the same pair of vertices and does not have loops. This basic structure is foundational in graph theory, as it enables many results and proofs to be applied without the complications introduced by multiple connections or self-connections, which are present in more general graph models.

Planar Graph

A planar graph is one that can be drawn on a plane without any of its edges crossing each other, except at their endpoints. This property is significant in both theoretical and practical applications since it relates to graph embeddings in the plane and is tied to various important results like Euler’s formula for planar graphs. The study of planar graphs also underpins many algorithmic techniques used in network visualization and circuit design.

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