Question

Prove that any Eulerian graph can be decomposed into a set of pairwise edge- disjoint cycles.

Short answer

An Eulerian graph is decomposed into edge-disjoint cycles by incrementally removing cycles.

Step-by-step solution

Understanding Eulerian Graphs

An Eulerian graph is a graph in which there exists a closed walk that visits every edge exactly once. A connected graph is Eulerian if and only if every vertex has an even degree.

Identify the Key Property

The key property of Eulerian graphs is that each vertex has an even degree. This ensures that each vertex can be the start and end of an even number of edges, forming cycles.

Starting the Decomposition

Begin at any vertex and traverse through the graph forming a cycle. Since the graph is Eulerian, it is always possible to return to the starting vertex without retracing any edge.

Remove the Cycle

Once a cycle is formed, remove all the edges of this cycle from the graph. This will still leave a graph in which all vertices in the cycle maintain even degrees because removing two edges (the ins and outs) keeps the degrees even.

Repeat the Process

Continue selecting cycles in the remaining graph, removing them after each identification. Since the graph originally had all even-degree vertices, the process continues until all edges are exhausted and a set of edge-disjoint cycles is fully constructed.

Finish the Decomposition

Repeat Steps 3-5 until no edges remain. Each step ensures edge-disjoint cycles are formed, covering all edges of the initial Eulerian graph once.

Key concepts

Graph Theory

Graph theory is a fascinating area of mathematics and computer science devoted to the study of graphs, or networks. A graph consists of vertices (or nodes) connected by edges. It's a way to model pairwise relationships between objects. Graphs can be either directed or undirected, depending on whether the edges have a direction. In essence, graph theory provides a powerful framework for solving problems related to network structures, such as social networks, computer networks, or even biological networks.
In the context of Eulerian graphs, graph theory explores the properties and behaviors of graphs that allow us to traverse every edge exactly once, forming an Eulerian circuit. This involves understanding the relationships and degrees of vertices, as these are crucial in determining whether a graph can be Eulerian.

Cycle Decomposition

Cycle decomposition refers to the process of breaking down a graph into cycles, which are paths that start and end at the same vertex, with no repeating edges or vertices, except for the starting/ending vertex. For Eulerian graphs, this concept is particularly important. Since these graphs have cycles covering every edge, breaking them into smaller cycles, which are edge-disjoint, provides insights into their structure.
By arranging a graph's edges into these cycles, each pairwise edge-disjoint, cycle decomposition ensures that all edges participate in one cycle only. This process aids in visualizing and understanding the overall shape and form of the Eulerian graph structure in simpler, manageable pieces.

Edge-Disjoint Cycles

In the context of Eulerian graphs, edge-disjoint cycles are essential. These are cycles in which no two share a common edge. Imagine them as distinct loops within a graph that do not overlap in terms of their edges.
The importance of edge-disjoint cycles lies in their ability to help decompose an Eulerian graph effectively. By ensuring each cycle uses different edges, you can break an Eulerian graph down into its simplest loop components without losing any information or edges in the graph. This division into separate, non-overlapping loops is what makes the decomposition possible and practical.

Eulerian Path

An Eulerian path is a trail in a graph that visits every edge exactly once. However, unlike an Eulerian circuit, it doesn’t necessarily start and end at the same vertex. The hunt for such a path is a central problem in graph theory called the Eulerian path problem.
A graph can host an Eulerian path if it has exactly zero or two vertices of odd degree. This criteria slightly loosens as compared to Eulerian circuits that demand no odd-degree vertices. Understanding Eulerian paths is crucial for breaking down Eulerian graphs and finding those edge-disjoint cycles that serve as the building blocks of the graph's structure.

Graph Degree

The degree of a vertex in a graph refers to the number of edges connected to it. Notably, a graph is Eulerian if each of its vertices has an even degree. This even degree condition ensures that, when traversing the graph, you enter and exit each vertex the same number of times, allowing for the formation of cycles.
In the cycle decomposition process, maintaining an even degree for all vertices is what permits removing cycles and still preserving the graph’s Eulerian qualities. This property flows directly from the definition of Eulerian graphs, supporting the decomposition into edge-disjoint cycles. It's an elegant principle ensuring balance and symmetry in the traversal of the graph, aiding both understanding and computational efficiency.