What makes an Eulerian trail?
An Eulerian trail is a specific type of path within a graph that traverses every single edge exactly once. Unlike an Eulerian circuit, which begins and ends at the same vertex, an Eulerian trail starts and ends at different vertices. The key characteristic that defines the existence of an Eulerian trail lies in the degrees of the vertices within the graph.
For a graph to possess an Eulerian trail, it must be connected (excluding isolated vertices) and have at most two vertices with an odd degree. All other vertices in the graph must have an even degree. The two vertices with odd degrees will serve as the start and end points of the Eulerian trail. If there are no vertices with an odd degree, then the graph actually contains an Eulerian circuit, which is a special case of an Eulerian trail where the start and end points are the same.
Conversely, if a connected graph has more than two vertices with an odd degree, it is impossible to construct an Eulerian trail. This is because each time a trail enters and leaves a vertex, it uses two edges, thus contributing an even number to the vertex’s degree. To traverse every edge exactly once, any vertex that is not the start or end point must have an even degree. The start and end points are the only exceptions, as they only have one “entry” or “exit” edge that isn’t paired with another.
What are the requirements for a Euler trail?
The existence of an Eulerian trail within a graph hinges on specific properties of its vertices. A trail, by definition, can revisit vertices but not edges. For a graph to contain an Eulerian trail, it must be connected, meaning there’s a path between any two of its vertices. Beyond connectivity, the key lies in the degree of each vertex, which is the number of edges connected to it.
Specifically, for an undirected graph to have an Eulerian trail, at most two vertices can have an odd degree. All other vertices must have an even degree. If there are exactly two vertices with an odd degree, the trail must start at one of these odd-degree vertices and end at the other. If all vertices have an even degree, then an Eulerian trail exists that can start and end at any vertex, forming an Eulerian circuit.
For a directed graph to possess an Eulerian trail, the requirements are slightly different. Again, the graph must be connected. The crucial condition here involves the in-degree (number of edges pointing to a vertex) and out-degree (number of edges pointing away from a vertex) of each vertex. An Eulerian trail exists if, for all but at most two vertices, the in-degree equals the out-degree. If there are exactly two such vertices, one must have an out-degree that is one greater than its in-degree (the start vertex), and the other must have an in-degree that is one greater than its out-degree (the end vertex). If for all vertices, the in-degree equals the out-degree, then an Eulerian trail (and circuit) exists.
What is the difference between a trail and a Euler trail?
Understanding the distinctions between a trail and an Euler trail is fundamental in graph theory. A trail in a graph is a walk that traverses distinct edges. This means that while vertices can be revisited, each edge in the sequence can only be used once. The length of a trail is determined by the number of edges it contains.
An Euler trail, on the other hand, is a very specific type of trail. It is a trail that visits every single edge of the graph exactly once. For an Euler trail to exist in a connected graph, there must be either zero or exactly two vertices with an odd degree. If there are two such vertices, the Euler trail must start at one of them and end at the other. If all vertices have an even degree, then the Euler trail is also an Euler circuit, starting and ending at the same vertex.
The key difference lies in the scope of edge traversal. While any sequence of distinct edges constitutes a trail, an Euler trail is a trail that achieves the remarkable feat of covering every edge in the entire graph. Therefore, every Euler trail is inherently a trail, but not every trail is an Euler trail.
How to do an Euler trail?
To successfully traverse an Euler trail, the fundamental principle is to use every edge of the graph exactly once. This means you cannot retrace your steps over an edge you’ve already used, nor can you leave any edge untouched. The process typically begins at a vertex and involves moving from one vertex to another by following an available edge. As you move, you mentally or physically mark the edge as “used.” The goal is to continue this process until all edges in the graph have been traversed.
The existence of an Euler trail is predicated on specific graph properties. For an undirected graph, an Euler trail exists if and only if it is connected and has at most two vertices of odd degree. If there are no vertices of odd degree, the trail is an Euler circuit, meaning it starts and ends at the same vertex. If there are exactly two vertices of odd degree, the trail must start at one of these odd-degree vertices and end at the other. When constructing the trail, prioritize paths that don’t immediately isolate a section of the graph, unless it’s the only remaining option to traverse a specific set of edges.
Practical Steps for Finding an Euler Trail:
* Identify Odd-Degree Vertices: Count the number of edges connected to each vertex. Vertices with an odd number of connected edges are “odd-degree” vertices.
* Start Point Selection:
* If there are no odd-degree vertices, you can start at any vertex.
* If there are exactly two odd-degree vertices, you must start at one of them.
* Edge Traversal Strategy:
* Always choose an edge that has not yet been traversed.
* Avoid choosing an edge that would disconnect the remaining un-traversed edges from the current component, unless it is the only remaining option to traverse a specific set of edges. This often means trying to leave “bridges” for later if other paths are available.
* Continue traversing edges until all edges have been used exactly once.