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Chapter 11: Problem 2

Show that when any edge is removed from \(K_{5}\), the resulting subgraph is planar. Is this true for the graph \(K_{3,3}\) ?

Short Answer

Expert verified
When any edge is removed from \(K_{5}\), the resulting subgraph is indeed planar, it can be drawn in a plane without edges crossing. However, this is not the case with the \(K_{3,3}\) graph. Even after removing an edge from \(K_{3,3}\), the graph remains non-planar.

Step by step solution

01

Create a Subgraph of \(K_{5}\) by Removing an Edge

Choose any edge of the \(K_{5}\) graph and remove it. As a result get a \(K_{5}\) graph minus one edge, which comprises 5 vertices and 9 edges.
02

Draw the Subgraph of \(K_{5}\) in the Plane

Now, try drawing the resulting graph in the plane such that none of the edges intersect except at the vertices. It is possible to do so because removing an edge destroys the property of \(K_{5}\) that makes it non-planar, hence the graph becomes planar.
03

Create a Subgraph of \(K_{3,3}\) by Removing an Edge

Choose any edge of the \(K_{3,3}\) graph and remove it. Get a \(K_{3,3}\) graph minus one edge, which comprises 6 vertices and 8 edges.
04

Draw the Subgraph of \(K_{3,3}\) in the Plane

Try to draw the resulting graph in the plane such that none of the edges intersect except at the vertices. This, however, is impossible. Despite the fact that we removed an edge, the subgraph of \(K_{3,3}\) without an edge is still non-planar, as it still contains a homeomorphic copy of \(K_{3,3}\).

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Most popular questions from this chapter

Give an example of a connected graph \(G\) where removing any edge of \(G\) results in a disconnected graph.

For \(n \in \mathbf{Z}^{+}\)where \(n \geq 4\), let \(V^{\prime}=\left\\{v_{1}, v_{2}, v_{3}, \ldots, v_{n-1}\right\\}\) be the vertex set for the complete graph \(K_{n-1}\). Construct the loop-free undirected graph \(H_{n}=(V, E)\) from \(K_{n-1}\) as follows: \(V=V^{\prime} \cup\\{v\\}\), and \(E\) consists of all the edges in \(K_{n-1}\) together with the new edge \(\left\\{v, v_{1}\right\\}\) a) Show that \(H_{n}\) has a Hamilton path but no Hamilton cycle. b) How large is the edge set \(E\) ?

a) Find the number of edges in \(Q_{8}\). b) Find the maximum distance between pairs of vertices in \(Q_{8}\). Give an example of one such pair that achieves this distance. c) Find the length of a longest path in \(Q_{8}\).

Prove that for \(n \geq 2\), the hypercube \(Q_{n}\) has a Hamilton cycle.

Seven towns \(a, b, c, d, e, f\), and \(g\) are connected by a system of highways as follows: (1) I-22 goes from \(a\) to \(c\), passing through \(b ;\) (2) I-33 goes from \(c\) to \(d\) and then passes through \(b\) as it continues to \(f ;(3)\) I-44 goes from \(d\) through \(e\) to \(a ;\) (4) \(\mathrm{I}-55\) goes from \(f\) to \(b\), passing through \(g\); and (5) I-66 goes from \(g\). to \(d\). a) Using vertices for towns and directed edges for segments of highways between towns, draw a directed graph that models this situation. b) List the paths from \(g\) to \(a\). c) What is the smallest number of highway segments that would have to be closed down in order for travel from \(b\) to \(d\) to be disrupted? d) Is it possible to leave town \(c\) and return there, visiting each of the other towns only once? e) What is the answer to part (d) if we are not required to return to \(c ?\) f) Is it possible to start at some town and drive over each of these highways exactly once? (You are allowed to visit a town more than once, and you need not return to the town from which you started.)
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