Exercise · ZW-HAMILTONIAN-CYCLE-009 · ★★★ Advanced
A bipartite graph has parts of 4 and 5 vertices. Can it have a Hamiltonian cycle?
Choose one option.
Show answer
- C. No: the cycle alternates between parts and forces them to be equal in size
Explanation
Cycles in a bipartite graph alternate parts, so they use equally many of each. The claim “9 is odd and cycles are even” is false: a triangle is an odd cycle (though it is not bipartite).
Review the concept: Hamiltonian cycle
A Hamiltonian cycle of a graph \(G\) with \(n\ge 3\) vertices is a cycle that visits every vertex of \(G\) exactly once.
\[\deg(v)\ge \tfrac n2\ \forall v\ \Longrightarrow\ \text{Hamiltonian cycle exists}\]
More practice problems: hamiltonian cycle
- ZW-HAMILTONIAN-CYCLE-001 How many distinct Hamiltonian cycles does K_5 have, not counting the starting vertex or direction?
- ZW-HAMILTONIAN-CYCLE-002 Which of these graphs has a Hamiltonian cycle?
- ZW-HAMILTONIAN-CYCLE-003 Which of these complete bipartite graphs has a Hamiltonian cycle?
- ZW-HAMILTONIAN-CYCLE-004 A simple graph has 8 vertices and each has degree at least 4. What can be guaranteed?
- ZW-HAMILTONIAN-CYCLE-005 A Hamiltonian cycle of a graph with 9 vertices uses how many edges?
- ZW-HAMILTONIAN-CYCLE-006 The bowtie is two triangles sharing one vertex (5 vertices). Which is true?
- ZW-HAMILTONIAN-CYCLE-007 Which condition prevents a graph from having a Hamiltonian cycle?
- ZW-HAMILTONIAN-CYCLE-008 In the cube Q_3 (vertices: 3-bit strings; edges: strings differing in one bit), which list is a Ham…