Theorem: Given a graph G has a Euler Circuit, then every vertex of G has a even degree
Proof: We must show that for an arbitrary vertex v of G, v has a positive even degree.
What does it mean by every even degree? When I think of an even degree I think of polynomial functions.
What I am trying to prove?
An Eulerian circuit is a traversal of all the edges of a simple graph once and only once, staring at one vertex and ending at the same vertex. You can repeat vertices as many times as you want, but you can never repeat an edge once it is traversed.
The degree of a vertex is the number of edges incident with that vertex.
So let $G$ be a graph that has an Eulerian circuit. Every time we arrive at a vertex during our traversal of $G$, we enter via one edge and exit via another. Thus there must be an even number of edges at every vertex. Therefore, every vertex of $G$ has even degree.