# Is there a non-planar, non-hamiltonian and eulerian graph?

I'm trying to find a graph that is non-planar, non-hamiltonian and eulerian but I can't find anyone. Is this possible?

Thanks

## 1 Answer

Take two copies of $$K_5$$ and identify one of the nodes in one with one of the nodes in the other.

• But K5 is 4-regular. Eulerian graphs has even degrees. If I join two vertex, this will have odd degree. – emee Dec 10 '18 at 0:55
• @emee: i'm not saying to connect the two vertices with an edge. I'm saying to combine them into one vertex. The combined vertex will have degree 8. – Henning Makholm Dec 10 '18 at 0:57
• Sorry but I don't undestand. d(v) + d(u) = 2.8 > 5 [Ore's theorem] – emee Dec 12 '18 at 22:24
• @emee: What is $v$ and $u$? The condition in Ore's theorem talks about pairs of distinct vertices. The graph I describe has eight vertices of degree 4 and one vertex of degree 8, so how do you get $2\cdot 8$ for even one pair of vertices? – Henning Makholm Dec 12 '18 at 22:44
• I get it. Thanks so much. I was drawing another graph – emee Dec 12 '18 at 22:47