Question

Let \(G\) be a connected graph with average degree greater than two. Prove that \(G\) contains at least two cycles.

Short answer

Graph \(G\) must contain at least two cycles as the number of edges exceeds the number of vertices by more than one.

Step-by-step solution

Understand the Problem Statement

We have a connected graph \(G\) where each vertex has an average degree greater than 2. This means the total degree of all vertices divided by the number of vertices is greater than 2. We need to prove that such a graph contains at least two cycles.

Express Average Degree Mathematically

If \(|V|\) is the number of vertices and \(|E|\) is the number of edges in \(G\), the average degree \( \bar{d} \) is given by \( \frac{2|E|}{|V|} \). Given that \( \bar{d} > 2 \), this simplifies to \( \frac{2|E|}{|V|} > 2 \).

Derive the Inequality

From \( \frac{2|E|}{|V|} > 2 \), multiply through by \(|V|\) to obtain \(2|E| > 2|V|\). Simplify this to get \(|E| > |V|\). This implies that there are more edges than vertices.

Recall Basic Graph Properties

A tree is a connected acyclic graph with \(|V| - 1\) edges. Any connected graph with edges more than \(|V| - 1\) must contain at least one cycle. Therefore, since \(|E| > |V|\), \(G\) must have at least one cycle.

Argue for the Existence of a Second Cycle

Since \(|E| > |V|\), the extra edge implies the presence of at least one cycle. Remove all edges of one cycle, leaving more than \(|V| - 2\) edges. This new graph, being a connected component with at least \(|V|\) vertices and more than \(|V| - 1\) edges, also contains a cycle.

Conclusion: State the Result

Since removing edges to eliminate one cycle still leaves a connected graph with an excess of edges over vertices, \(G\) must contain at least two cycles.

Key concepts

Connected Graphs

In graph theory, a connected graph is one where there is a path between any two vertices. This connectivity means that the graph is all in one piece; there are no isolated parts or lone vertices. This is an important concept because it ensures the graph has a certain level of completeness and complexity.

In a connected graph, if you start at one vertex, you can travel along the edges and reach any other vertex. This property is key when proving certain traits of a graph, such as the ability to form cycles. Connected graphs are foundational structures that allow for the exploration of more complex properties, like average degree and cycle formation.

• All vertices are reachable from any other vertex.
• Connected graphs can have cycles, but they can also be trees, which are acyclic.
• The complete connection plays a vital role in determining other graph properties.

Average Degree

The concept of average degree in a graph helps us understand how connected the vertices are on average. It is calculated by taking the sum of the degrees of all vertices and dividing by the number of vertices. For a graph to have an average degree greater than 2 means on average, each vertex is part of more than two edges.

The formula for average degree is expressed as \( \bar{d} = \frac{2|E|}{|V|} \), where \(|E|\) is the number of edges and \(|V|\) is the number of vertices. In simpler terms, this calculation gives insight into the graph's density: the more edges compared to vertices, the higher the average degree.

• A higher average degree typically implies more complexity.
• It reflects how tightly knitted the graph is.
• If average degree is greater than 2, like in our problem, it hints towards the presence of cycles.

Cycles in Graphs

Cycles in graphs are paths that start and end at the same vertex, without traversing any edge more than once. Finding cycles is essential in graph theory for understanding the graph's structure and behavior. In a graph with more edges than vertices, cycles are often inevitable.

A tree, being an acyclic connected graph, has exactly \(|V| - 1\) edges. Any additional edges would inevitably form at least one cycle. Therefore, if a graph has even just one more edge than vertices, it must contain a cycle. This understanding is crucial in the given exercise, where our graph's high average degree suggests an excess of edges over vertices.

• A cycle is formed when there is more than \(|V| - 1\) edges in a connected graph.
• With an average degree greater than 2, the graph surpasses the minimum edge requirement for forming cycles.
• After removing one cycle, a graph conclusively still supports a second cycle owing to excessive edge count.