- Background Information:
I am studying graph theory in discrete mathematics. As I was reading my notes, I came across few definitions. I think I have noticed a pattern, but I need to confirm it with someone, thanks.
- Definition of Euler Graph:
Let G = (V, E), be a connected undirected graph (or multigraph) with no isolated vertices. Then G is Eulerian if and only if every vertex of G has an even degree.
- Definition of Euler Trail:
Let G = (V, E), be a conned undirected graph (or multigraph) with no isolated vertices. Then G contains a Euler trail if and only if exactly two vertices of G are of odd degree.
- My Example:
- Look at the image above, consider the vertices with black edges (imagine blue edge does not exist) to be undirected graph G.
- Questions & Patterns:
Without the blue edge (a,b), we have a Euler trail, am I right? Explain why.
With the blue edge (a,b), we have a Euler graph which is also a Euler cycle in this case, am I right? Explain why.