The Fundamental Theorem of Graph Theory often refers to Euler's Theorem for connected graphs, which provides a criterion for when a graph contains an Eulerian circuit. An Eulerian circuit is a circuit that visits every edge of a graph exactly once and returns to the starting vertex.

Euler's Theorem (Fundamental Theorem of Graph Theory)

For a connected graph, the graph contains an Eulerian circuit if and only if every vertex has an even degree.

Statement:

Let $G = (V, E)$ be a connected graph. The graph $G$ has an Eulerian circuit if and only if:

1. $G$ is connected.
2. Every vertex in $G$ has an even degree.

Proof:

Necessity (if the graph has an Eulerian circuit, then every vertex has even degree):

1. Consider a graph $G$ with an Eulerian circuit.
2. As the Eulerian circuit passes through any vertex $v$, each time it visits $v$, it must both enter and leave along an edge.
3. Hence, the number of edges incident to any vertex $v$ (its degree) must be even since the number of entrances must equal the number of exits.
4. Thus, for every vertex in $G$, the degree must be even.

Sufficiency (if every vertex has even degree and the graph is connected, then it has an Eulerian circuit):

1. Assume $G$ is a connected graph and all vertices have even degree.
2. Start at any vertex $v$ and begin traversing edges, marking them as used.
3. Since every vertex has even degree, whenever you enter a vertex, there is always an unused edge to leave by (except possibly at the very end, when you return to the starting vertex).
4. This allows you to traverse all edges without getting stuck at any vertex except the starting vertex, where you will eventually return.
5. Therefore, an Eulerian circuit exists.

Thus, the graph contains an Eulerian circuit if and only if every vertex has even degree and the graph is connected.

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Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. What is an Eulerian path, and how does it differ from an Eulerian circuit?
2. Can a graph have an Eulerian path if some vertices have odd degrees?
3. How does the Fundamental Theorem apply to directed graphs?
4. How can the theorem be extended to graphs with multiple disconnected components?
5. What is the relationship between Eulerian circuits and Hamiltonian circuits?

Tip: When checking for Eulerian circuits, always verify the degree of each vertex—this can save time when determining if a graph has such a circuit.