No, this is an incorrect definition of a Euler cycle.
Use the rule of product: first select a vertex in one set and then a vertex in the other set to which it is connected.
Each vertex has n - 1 adjacent edges. This counts each edge twice.
Hello
I have a couple questions about graphs that I am not sure about...
1. If a graph contains a cycle that includes all the edges, the cycle is an Euler cycle. My answer: Yes, by the definition of the Euler cycle - a cycle in a graph G that includes
all of the edges and all of the vertices of G is called an Euler cycle.
2. Find a formula for the number of edges in Km,n . Answer: m*n. But not sure how to prove that...
3.Find a formula for the number of edges in Kn . Answer: (n(n-1))/2. But not sure how to prove that...
Thank you.
No, this is an incorrect definition of a Euler cycle.
Use the rule of product: first select a vertex in one set and then a vertex in the other set to which it is connected.
Each vertex has n - 1 adjacent edges. This counts each edge twice.
OK, my bad. Apparently, the definition of a cycle in your book requires that vertices are not repeated. Wikipedia says this here: "Like path, this term [cycle] traditionally referred to any closed walk, but now is usually understood to be simple [i.e., no vertices (and thus no edges) are repeated] by definition." The reason I objected was that Euler cycle includes each edge exactly once. But if a cycle includes any edge no more than once by definition, then it is possible to say that Euler cycle is a cycle that contains all edges.