Exam 10: A: Graphs

How many different channels are needed for six television stations (A, B, C, D, E, F ) whose distances (in miles) from each other are shown in the following table? Assume that two stations cannot use the same channel when they are within 150 miles of each other? A B C D E F A - 85 175 100 50 100 B 85 - 125 175 100 130 C 175 125 - 100 200 250 D 100 175 100 - 210 220 E 50 100 200 210 - 100 F 100 130 250 220 100 -

either give an example or prove that there are none. -A graph with region-chromatic number equal to 6.

either give an example or prove that there are none. -A simple digraph with indegrees 0, 1, 1, 2 and outdegrees 0, 1, 1, 1.

Give a recurrence relation for e_n = the number of edges of the graph $K _ { n }$

fill in the blanks. -There are ____ non-isomorphic simple digraphs with 3 vertices and 2 edges.

The Math Department has 6 committees that meet once a month. How many different meeting times must be used to guarantee that no one is scheduled to be at 2 meetings at the same time, if committees and their members are: $C _ { 1 } = \{$ Allen, Brooks, Marg $\} , C _ { 2 } = \{$ Brooks, Jones, Morton $\} , C _ { 3 } = \{$ Allen, Marg, Morton $\}$ , $C _ { 4 } = \{$ Jones, Marg, Morton $\} , C _ { 5 } = \{$ Allen, Brooks $\} , C _ { 6 } = \{$ Brooks, Marg, Morton $\}$ .

Draw a cubic graph with 6 vertices that is not isomorphic to K_3,3, or else prove that there are none.

fill in the blanks. -The length of the longest simple circuit in $W _ { 10 }$ is ____.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for W₅.

Determine whether this graph is planar.

Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path?

fill in the blanks. -The region-chromatic number for $W _ {9 }$ = ____.

In K_3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 3.

either give an example or prove that there are none. -A planar graph with 7 vertices, 9 edges, and 5 regions.

Many supermarkets use loyalty or discount cards to keep track of who buys which items. How can graphs be used to model this relationship? Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed? Does this graph have any special properties?

Draw a cubic graph with 8 edges, or else prove that there are none.

Are these two graphs isomorphic?

Determine whether this graph is planar.

Determine whether the graph is strongly connected, and if not, whether it is weakly connected.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for Q₃.