Mathematics — Graph Theory · digraph-cycles-degrees
Problem
Let D be a multidigraph with at least one vertex. Prove the following:
(a) If each vertex v of D satisfies deg+v>0, then D has a cycle.
(b) If each vertex v of D satisfies deg+v=deg−v=1, then each vertex of D belongs to exactly one cycle of D. Here, two cycles are considered to be identical if one can be obtained from the other by cyclic rotation.