- Is there a simple graph of 9 vertices with degree sequence 3, 3, 3, 3, 5, 6, 6, 6, 6?
- Is there a bipartite graph of 8 vertices with degrees 3, 3, 3, 5, 6, 6, 6, 6?
- In a simple graph with at least two vertices, show that there are at least two vertices with the same degree.
- A directed graph (or digraph) is a graph X together with a function assigning to each edge, an ordered pair of vertices. The first vertex is called the tail of the edge and the second is called the head. To each vertex, v, we let
be the number of the edges for which v is the tail and
the number for which it is the head. We call
the outdegree and
the indegree of v. Prove that
where the sum is over the vertex set of X.
- In any digraph, we define a walk as a sequence
with
the tail of
and
its head. The analogous notions of trail, path, circuit, and cycle are easily extended to digraphs in the obvious way. If X is a digraph such that the outdegree of every vertex is at least one, show that X contains a cycle.
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Nalin Pithwa