Exercise · ZW-PLANAR-GRAPH-008 · ★★☆ Intermediate
A connected planar graph with no triangles satisfies \(e\le 2v-4\). Is \(K_{3,3}\) (6 vertices, 9 edges, no triangles) planar?
Choose one option.
Show answer
- A. No, because \(9>2\cdot6-4=8\)
Explanation
If \(K_{3,3}\) were planar it would have \(e\le 8\), but it has 9. The bound \(3v-6=12\) decides nothing here, since it is only a necessary condition.
Review the concept: Planar graph
A planar graph is a graph that can be drawn in the plane with no crossing edges; such a drawing, for \(v\) vertices and \(e\) edges, splits the plane into \(f\) faces.
\[v-e+f=2\]
More practice problems: planar graph
- ZW-PLANAR-GRAPH-001 Which of the following complete graphs is planar?
- ZW-PLANAR-GRAPH-002 A connected planar graph has v=8 vertices and e=12 edges. How many faces (counting the outer one) d…
- ZW-PLANAR-GRAPH-003 A plane drawing of a connected graph has 6 vertices and 5 faces. How many edges does it have?
- ZW-PLANAR-GRAPH-004 A simple, connected, planar graph has v=7 vertices. What is the largest possible number of edges?
- ZW-PLANAR-GRAPH-005 Why is K_5 not planar?
- ZW-PLANAR-GRAPH-006 A tree with 10 vertices is drawn in the plane. How many faces does the drawing have?
- ZW-PLANAR-GRAPH-007 A planar graph, not necessarily connected, has 2 connected components, v=9 vertices and e=10 edges.…
- ZW-PLANAR-GRAPH-009 A maximal planar graph with v=10 vertices (every face a triangle) has e=24 edges. How many faces do…