6.1. Basic definitions and theorems
A circuit in a graph G containing all the edges is said to be an Euler circuit of G.
An Euler trail contains all edges but it is not closed.
A graph is Eulerian if it has an Euler circuit.
A graph G is defined to be even (odd) graph if all its vertices have even (odd) degree.
Theorem 6.1. A non-trivial connected graph has an Euler circuit iff each vertex has even degree.

are the only vertices of odd degree.
Theorem 6.3. If G is a connected graph
with 2k odd vertices ,
then
can be partitioned into subsets
so that for each
is a trail connecting odd vertices and such
that at most one of these trails has odd length.
Theorem 6.4. A nontrivial connected graph G
is eulerian iff can be partitioned into subsets
, where each subgraph
is a cycle.