HOMEWORK GRAPH THEORY

Matching Question Match the condition on the left with the existence statement on the right. 1. A connected multigraph with at least two vertices where all vertices have even degree 2. A connected multigraph with exactly two vertices of odd degree 3. A connected multigraph with exactly four vertices of odd degree

1. The graph has an Euler circuit. 2. The graph has an Euler path but not an Euler circuit. 3. The graph has neither an Euler circuit nor an Euler path.

Multiple Choice Question What is the chromatic number of this graph?

4

Multiple Select Question Select all that apply Which of these statements about Hamilton circuits are true?

A simple graph with n vertices must have a Hamilton circuit if n ≥ 3 where deg(u) + deg(v) ≥ n whenever u and v are nonadjacent vertices. A simple graph with n vertices must have a Hamilton circuit if n ≥ 3 where the degree of every vertex is at least n / 2.

Fill in the Blank Question A simple path in a graph G passing through all vertices in the graph exactly once is called a (BLANK 1) path.

Blank 1: Hamilton or Hamiltonian

Fill in the Blank Question A coloring of a simple graph is the assignment of a color to each vertex so that no two adjacent vertices are assigned the same color. The (BLANK 1) number of a graph is the least number of colors needed for a coloring of the graph.

Blank 1: chromatic

Fill in the Blank Question The operation on a simple graph of removing an edge {u, v} and then adding a new vertex w together with new edges {u, w} and {w, v} is called (BLANK 1) subdivision. Two graphs are called (BLANK 2) if they can be obtained from the same graph by a sequence of such operations.

Blank 1: elementary Blank 2: homeomorphic

Question Mode Fill in the Blank Question A graph is called (BLANK 1) if it can be drawn in the plane without any edges crossing.

Blank 1: planar

Question Mode Multiple Choice Question Which of these criteria can be used to show that K3,3 is nonplanar?

If a connected planar simple graph has e edges and v vertices, where v ≥ 3, and no circuits of length 3, then e ≤ 2v - 4.

Question Mode Multiple Select Question Select all that apply Which of these problems is solved by determining whether the graph that is plainly implied by the problem is planar or not?

Designing a road network that can be built without using underpasses or overpasses Determining whether we can connect utilities to house at ground level without any connections crossing Determining whether we can build an electronic circuit so that none of the connections of transistors cross

Question Mode Multiple Select Question Select all that apply Which of these are applications of graph coloring?

Determining the number of index registers needed to speed up execution of loops Scheduling exams so that no student has two exams at the same time Assigning channels to broadcast stations so that no two stations are too close together

Multiple Select Question Select all that apply Which of these statements are true about the dual graph of a planar map?

Each region of the map is represented by a vertex. Two vertices of the dual graph are connected by an edge if the regions they represent have a common border. The dual graph is planar.

Question Mode Multiple Select Question Select all that apply Which of these directed graphs have an Euler path? (Select all that apply.)

G1 G2

Question Mode Multiple Select Question Select all that apply Which of these are applications of Euler paths or circuits?

Finding a route for a postman so that the route covers each street in the route exactly once Finding a route for a snowplow that traverses each street in the route exactly once

Multiple Select Question Select all that apply Which of these problems involve finding the shortest path in a weighted graph?

Finding the airline route with the least total flight time between two cities with zero or more stops Finding the least expensive airline route with zero or more stops between two cities when the cost is the sum of the cost of the legs Finding the an airline route between two cities with the shortest total mileage Finding the total distance a traveling salesman must drive on a road network to travel between two cities

Multiple Select Question Select all that apply Kuratowski's theorem says that a graph is nonplanar if it contains a subgraph homeomorphic to which of these graphs? (Select all that apply.)

K5 K3,3

Question Mode Multiple Select Question Select all that apply Which of these are applications of Hamilton circuits or paths?

Minimizing the size of errors when analog signals are converted to digital through the use of Gray codes Finding the ways a traveling salesman can visit all the cities he services Finding a way for a garbage truck to visit all customers without overlap

Question Mode Multiple Select Question Select all that apply Which of these statements about the four color theorem are true?

Nonplanar graphs can have arbitrarily large chromatic numbers. Many incorrect proofs were published in the more than 120 years between when the theorem was conjectured and when it was finally proved. The four color theorem states that the chromatic number of a planar graph does not exceed four. The first proof of the four color theorem relied on computer computations using over 1000 hours of computing time.

Multiple Choice Question Which of these provides a correct analysis of the famous Köningsberg bridge problem?

The bridges multigraph has four odd vertices, so it has no Euler circuit. Thus one cannot cross each bridge exactly once and return to the start.

Question Mode Multiple Select Question Select all that apply Which of these statements about the chromatic number are true? (Select all that apply.)

The chromatic number of the wheel Wn when n ≥ 3 is 3 if n is even and 4 if n is odd. The chromatic number of the complete bipartite graph Km,n is the same for all m ≥ 1, n ≥ 1.

Question Mode Multiple Select Question Select all that apply Which of these statements about Hamilton paths and Hamilton circuits are true?

The complete bipartite graph Kn,n+1 has a Hamilton path for n≥3. A graph with a pendant vertex cannot have a Hamilton circuit. A Hamilton circuit cannot contain a smaller circuit within it.

Question Mode Multiple Select Question Select all that apply Which of these statements about planar graphs are true?

There are 10 regions in a planar representation of a graph with 20 edges and 12 vertices. A connected planar simple graph with 15 vertices and 16 edges could be planar.

Question Mode Multiple Select Question Select all that apply Which of these graphs have a Hamilton circuit?

W4 K5 C5

Multiple Select Question Select all that apply Which of these graphs are planar?

W5 C5 Q3

Multiple Choice Question What is the name for a graph that has a number assigned to each edge?

Weighted graph

Multiple Choice Question Which of these relationships about the number of edges e, the number of vertices v, and the number of regions r in a planar graph is correct?

r = e - v + 2