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Euler's Theorem


Euler's Theorem describes a condition to which a connected graph $G = (V(G), E(G))$ is Eulerian. We will look at a few proofs leading up to Euler's theorem.

We will go about proving this theorem by proving the following lemma that will assist us later on.

\begin{align} V(P) = \left\{ x_1, x_2, ... , x_n \right\} \end{align} The image below gives a visual interpretation of the proof of Lemma 1 from above. Since the minimum degree $\delta (G) ≥ 2$, the vertices $x_2, x_3, ... x_{n-1}$ satisfy this by the definition of being a maximal path. But $x_1$ and $x_n$ must also satisfy this, so there must exist some other edges (gray) in the graph $G$ otherwise $\delta (G) = 1$.

To illustrate Lemma 2 let's look at the following graph: Notice that the degree of each vertex is even, so we can apply what the lemma from earlier to obtain some cycles of the graph by starting at a vertex and traversing along the graph until we return to the vertex. Clearly, no edge was ever repeated. We can easily see this by extracting the cycles individually: We are now ready to prove Euler's Theorem.