Question

Show that the edge set of a graph in which each vertex has even degree may be partitioned into edge sets of cycles of the graph.

Short answer

A graph with vertices having even degrees can be partitioned into cycles using its Eulerian circuits.

Step-by-step solution

Understanding the Problem

We need to show that a graph where each vertex has an even degree can have its edges partitioned into a set of cycles. A cycle is a path that starts and ends at the same vertex without repeating any edges.

Eulerian Graphs

A graph is Eulerian if it has a circuit that visits every edge exactly once and returns to the starting vertex. One of Euler’s Theorems states that a connected graph has an Eulerian circuit if and only if every vertex has an even degree.

Checking the Graph

Given that every vertex in the graph has an even degree, according to Euler’s Theorem, the graph will have an Eulerian circuit if it is connected. If the original graph is not connected, consider the components separately.

Decompose Eulerian Circuit

An Eulerian circuit is, by definition, a sequence of edges where every edge is visited exactly once. This sequence can be divided into cycles by choosing any edge as a starting point and tracing through the edges back to that point.

Partition the Edge Set

For each connected component of the graph with even-degree vertices, start from an Eulerian circuit and repeatedly break it into disjoint cycles. As every edge is part of one Eulerian circuit, all edges will be covered, completing the partition into cycles.

Key concepts

Eulerian Circuit

In graph theory, an Eulerian circuit is a trail in a graph that starts and ends at the same vertex and travels along each edge exactly once. It is named after the famous mathematician, Leonhard Euler, who laid the foundations of this concept. Euler's pioneering work in graph theory began with solving the problem known as the Seven Bridges of Königsberg.

To identify an Eulerian circuit in a graph, the graph must have two essential properties:

• Every vertex of the graph should have an even degree. This means each vertex connects to an even number of edges, allowing a smooth entry and exit path.
• The graph should be connected, meaning there should be a path between any two vertices.

Detecting an Eulerian circuit can assist in various applications like designing efficient pathways and network routing.

Even Degree

The term even degree refers to the number of edges incident to a vertex being an even number. If each vertex in a graph has an even degree, it implies that the graph is balanced in the context of entering and exiting paths.

Here's why even degrees matter:

• Facilitates an Eulerian circuit: If every vertex has an even degree, there's an intuitive pathway to return to the starting vertex without re-tracing any edge.
• Simplifies decomposition into cycles: Sources and destination nodes always match up, making it easier to partition the graph into isolated cycles.

Understanding vertex degrees is key to determining various properties of graphs, including the possibility of Eulerian circuits.

Cycle Decomposition

Cycle decomposition involves breaking down the graph into a set of cycles, which are closed paths that start and end at the same vertex without repeating edges. This process helps in understanding complex graph structures by simplifying them into fundamental components.

To perform cycle decomposition, follow these steps:

• Start from an Eulerian circuit, which is a large cycle covering all edges once.
• Select edges to form smaller, disjoint cycles until all are covered.

Cycle decomposition can be useful for various applications, like performing network optimizations and solving resource allocation problems efficiently.

Connected Graph

A connected graph is a type of graph where there is a path between every pair of vertices. This property is essential in ensuring that there is no vertex that is "isolated" from the rest of the graph.

Connected graphs are fundamental in graph theory for a few reasons:

• Basis for Eulerian circuits: Without a path between every pair of vertices, an Eulerian circuit is not possible even if every vertex has an even degree.
• Ensures continuity in circuits and paths: Connections between vertices allow seamless transitions from one point to another.

A graph being connected proves vital in applications ranging from network design to traffic routing, where continuity and traversal are critical.