So I'm basically failing my discrete mathematics class, with Graph Theory, because I don't know how to define something generally and not specifically, and all our teacher does is read the textbook and not answer our emails. Anyway I want to learn:
For Instance, One of our proofs is:
Let G be a C7 graph (A circuit graph with 7 vertices). Prove that G^C (G complement) has a Euler Cycle
Well I know that An Euler cycle is a cycle that contains all the edges in a graph (and visits each vertex at least once).
And obviously the complement of G would be all the same vertices, but not using any of the same edges and connecting all the ones that weren't connected.
Also its not mentioned, but I think were assuming that G has a Euler cycle and you just need to prove that G^C also has that Euler cycle. Of course I could do this specifically by mentioning a specific example, but I'm lost on how to write the proof.
You're asked to prove the existence of an Euler cycle in a specific graph. Just give one and you're done. This might be a bit tedious as we have a lot of edges.
You can also quote a theorem:
Which I hope has been covered in your course.
Well, in $C_7$ every vertex has degree 2 (out of the 6 connections they could have had), so in the complement of $C_7$ every vertex has degree $4$ (so all even). This complement is clearly connected (If the cycle is $1 \to 2 \to 3 \to \ldots \to 6 \to 7 \to 1$: the component of $1$ contains $\{1,3,4,5,6\}$ directly and next $1$ and $7$ in two steps via $3$ resp. $5$ and so all points are in one component) so the theorem applies and the complement of $C_7$ is Eulerian.