Euler Formula Quiz: Work with Complex Exponential Forms

Euler trails use every edge exactly once but may start and end at different vertices. An Euler trail (or Euler path) is designed to traverse every edge without repetition. Because the trail does not require returning to the starting point, the endpoints may differ, which distinguishes it from an Euler circuit.

A graph must be connected (ignoring isolated vertices) for an Euler trail or circuit to exist. If the graph falls into separate pieces, no single walk can cover all edges continuously.

The classical Euler characterization states that a connected graph can have an Euler trail only when precisely two vertices have odd degree; these become the start and end points of the traversal.

Odd degrees count toward the Euler condition. With vertices of degree 1 and 3, the graph has two odd vertices, allowing an Euler trail but preventing a circuit since circuits require zero odd vertices.

To return to the starting point without getting stuck, every visit to a vertex must allow leaving through a matching unused edge. This balance is only possible when every vertex has even degree.

In star-like structures, outer leaves each contribute an odd degree of 1, and the center contributes k. For there to be exactly two odd vertices, the center must also be odd, canceling all but two odd vertices.

The lemma states that each edge contributes two degree-counts—one to each of its endpoints. Thus, adding all vertex degrees always gives twice the number of edges, a fundamental counting fact.

If a connected graph has exactly two odd-degree vertices, Euler’s theorem guarantees that a trail exists which uses every edge exactly once, starting at one odd vertex and ending at the other.

Directed Euler circuit ↔ balanced in/out degrees. For directed graphs, each vertex must have the same number of incoming and outgoing edges to ensure that the walk can enter and leave equally, allowing a closed Euler circuit.

Each edge contributes 1 to indegree and 1 to outdegree.

In a directed graph, every edge must leave one vertex and enter another, so total indegree equals total outdegree across the whole graph, preserving global balance.

If a graph has more than two odd-degree vertices, it cannot support a single continuous walk that uses every edge exactly once, since additional odd vertices create unavoidable dead-ends.

Practical applications model tasks where every route (edge) must be covered once without unnecessary repetition—classic examples include garbage collection, mail delivery, and maintenance routes.

If removed edges do not disturb the even-degree structure or connectivity, the remaining graph may still satisfy Euler’s conditions, demonstrating how robust Eulerian properties can be.

For directed graphs, strong connectivity ensures all vertices are reachable, and matching indegree and outdegree at each vertex ensures balanced movement, together guaranteeing a directed Euler circuit.

Adding a single edge increases the degrees of exactly two vertices by 1 each, toggling their parity. Therefore, to correct four odd vertices (reducing them to zero), two edges must be added.

A Euler trail is a walk through a graph that uses every edge exactly once, possibly beginning and ending at different vertices. It generalizes the idea of a continuous walk that covers all edges.

A circuit is a special case of a trail. All Euler circuits are Euler trails because they satisfy the requirement of using every edge exactly once but with the additional constraint of starting and ending at the same vertex.

This is the core result of Euler’s work: a connected graph with exactly two odd-degree vertices supports a single continuous edge-covering path between them.

All vertices have degree 2. A graph where every vertex has degree 2 consists of disjoint cycles. If it is connected, it is a single cycle and thus automatically has an Euler circuit.

Exactly two odd vertices → Euler path but no circuit. With two odd vertices, the traversal must begin at one odd vertex and end at the other, making a circuit impossible since a circuit requires returning to the start.