Answer
There is no simple graph with four vertices of
degrees 1, 2, 3, and 4
Work Step by Step
Suppose there were a simple graph with four vertices of
degrees 1, 2, 3, and 4. Then the vertex of degree 4 would
have to be connected by edges to four distinct vertices other
than itself because of the assumption that the graph is simple
(and hence has no loops or parallel edges.) This contradicts
the assumption that the graph has four vertices in
total. Hence there is no simple graph with four vertices of
degrees 1, 2, 3, and 4.
