Consider the following statements regarding planar multigraphs. Determine whether each statement is True (T) or False (F).

1. If a connected planar multigraph is bipartite, its total number of edges E and vertices V must strictly satisfy the inequality E <= 2V - 4 for all V >= 3.

2. In any planar embedding of a connected multigraph, a bridge (cut-edge) in the primal graph corresponds to a self-loop in its geometric dual.

3. The sum of the degrees (lengths) of all faces in a planar embedding of a multigraph is exactly 2E, even if the graph contains self-loops or parallel edges.

4. Contracting any single edge in a non-planar multigraph will always yield a graph that is also non-planar.

5. A connected planar multigraph possesses an Eulerian circuit if and only if its geometric dual is a bipartite multigraph.

6. The Four Color Theorem guarantees that every planar multigraph can be properly vertex-colored using 4 or fewer colors.

Which of the following sequences correctly identifies the statements as True (T) or False (F)?

a) T, F, F, T, F, T

b) F, F, T, T, F, F

c) F, T, T, F, T, F

d) T, T, F, F, T, T

e) None of the above

Comments

  1. Joao MeidanisApril 27, 2026 at 12:18 PM

    Nice question, but it speaks of duals, contraction, and coloring, concepts we did not explore in the class.

    ReplyDelete

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