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. $$ \begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { then the thickness of } G \text { is }} \\ {\text { at least }[e /(3 v-6)] .}\end{array} $$

The thickness of a simple graph $G$ is the smallest number of planar subgraphs of $G$ that have $G$ as their union.
$$
\begin{array}{l}{\text { Show that if } G \text { is a connected simple graph with } v \text { ver- }} \\ {\text { tices and } e \text { edges, where } v \geq 3, \text { then the thickness of } G \text { is }} \\ {\text { at least }[e /(3 v-6)] .}\end{array}
$$

Key Concepts

Covering a Graph with Planar Subgraphs

Covering a graph with planar subgraphs involves partitioning the edge set of a graph into subsets so that each subset forms a planar graph. This concept is instrumental in computing the thickness of a graph, since the minimum number of such subsets needed gives the graph's thickness.

Edge Bound for Planar Graphs

One of the key consequences of Euler's formula is the edge bound for planar graphs, which states that a planar graph with at least three vertices has no more than 3V - 6 edges. This upper bound is critical when decomposing a non-planar graph into planar subgraphs, as it limits the complexity each subgraph can have.

Euler's Formula

Euler's formula is a fundamental theorem in planar graph theory, stating that for any connected planar graph, the number of vertices (V), minus the number of edges (E), plus the number of faces (F) equals 2 (i.e., V - E + F = 2). This formula is pivotal in deriving further properties and bounds for planar graphs.

Planar Graph

A planar graph is one that can be drawn in the plane without any of its edges crossing except at common endpoints. Planar graphs are central in graph theory because they satisfy certain properties and limitations, such as specific bounds on the number of edges relative to vertices.

Graph Thickness

Graph thickness is a measure of how 'non-planar' a graph is; it is defined as the minimum number of planar subgraphs needed such that their union forms the original graph. This concept helps in understanding and partitioning the complexity of a graph’s structure in terms of planarity.

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