COMP SCI 242 Discrete Mathematics II
Exam # 1
NAME
Spring 2009
F. Baulieu
1) [25 points] A cube has six "faces" (Front, Back, Right, Left, Top, Bottom) . We define a graph G as
follows: vertices are the "faces" of the cube, and two vertices are adjacent if the corresponding "faces"
of the cube share an edge of the cube. (I.e. the front and top of the cube share an edge).
Caution: The “faces” of the cube are different from the faces of a graph!
a) Draw the graph G.
b) Prove or disprove: G is bipartite.
c) Prove or disprove: G is planar.
d) Prove or disprove: G has an Euler cycle.
e) Prove or disprove: G is complete.
2) [10 points] Draw the digraph whose adjacency matrix is:
3) [10 points] In the blanks provided, mark each statement as True (T) or False (F):
a) If a graph has an Euler cycle, then it also has a Hamiltonian cycle.
b) If a graph is acyclic, then it is planar.
c) Every Tree has only one component.
d) Every Tree is bipartite.
e) An incidence matrix must be square .
4) [15 points] Use Dijkstra’s algorithm for the Single Source Shortest Pat Problem to find the
lengths of the shortest paths from A to each of the other vertices in the graph shown below:
Show your work!
A
B
3
5
4
D
7
E
F
2
6
4
3
1
2
1
G
C
2
H
I
5) [15 points] How many distinct (non-isomorphic) nonempty connected subgraphs of K2,2 are
there? Justify your answer.
6) [15 points] Find an Euler cycle for the graph G pictured below, or prove that no such cycle
exists.
A
B
C
D
E
F
7) [10 points] Find two distinct spanning trees for the graph in problem 7.