Exercise · ZW-EULERIAN-CYCLE-005 · ★★☆ Intermediate
A connected graph has exactly two vertices of odd degree. Which statement is true?
Choose one option.
Show answer
- B. It has no Eulerian cycle, but it has a trail that uses each edge once
Explanation
An open trail using every edge exists when there are exactly two odd vertices, starting and ending at them; but it cannot be closed into a cycle.
Review the concept: Eulerian cycle
An Eulerian cycle of a graph \(G=(V,E)\) is a cycle that traverses every edge of \(G\) exactly once.
\[G\text{ connected and }\deg(v)\text{ even }\forall v\in V\iff\text{Eulerian cycle exists}\]
More practice problems: eulerian cycle
- ZW-EULERIAN-CYCLE-001 Which of these graphs has an Eulerian cycle?
- ZW-EULERIAN-CYCLE-002 A connected graph has vertex degrees 2,4,4,6. Does it have an Eulerian cycle?
- ZW-EULERIAN-CYCLE-003 K_7 has an Eulerian cycle. How many edges does it traverse?
- ZW-EULERIAN-CYCLE-004 Which of these complete bipartite graphs has an Eulerian cycle?
- ZW-EULERIAN-CYCLE-006 A connected graph has six vertices with degrees 2,2,2,2,4,x. Which value of x allows an Eulerian cy…
- ZW-EULERIAN-CYCLE-007 If a graph has an Eulerian cycle, which statement is always true?
- ZW-EULERIAN-CYCLE-008 The Königsberg bridges graph has vertices of degrees 3,3,3,5. Why is there no Eulerian cycle?
- ZW-EULERIAN-CYCLE-009 From K_5 the 4 edges of a cycle 1-2-3-4-1 are removed. What graph remains?