Question

Draw a graph which has an Euler circuit but is not planar. Formalize the graph in the form 

Draw a graph which does not have an Euler path and is also not planar. Formalize the graph in the form 

Note: If you cannot draw the graph due to technical reasons, it is OK to just use formal notation and describe the graph textually.

Answer 1

A non-planar graph with Euler circuit is $K_5$ (it is a common result about non-planarity and the degree of each vertex is 4, which is even)

A graph without Euler path which is not planar is $K_5\cup\{x\}$ (with one additional point). It is not connected graph, hence it has no Euler path.