CPTS553 Assignment 3 Solved

Description

Questions with a (⋆) are each worth 1 bonus point for 453 students.

1. Let 𝑛 ≥ 3 be given and let 𝐶_𝑛 be the 𝑛-cycle whose vertices are {0,1,2,… , 𝑛 − 1}. Recall that 𝑎𝑏 is an edge if and only if 𝑎 = 𝑏 + 1 mod 𝑛 or 𝑏 = 𝑎 + 1 mod 𝑛.
   1. How many vertices does 𝐶_𝑛 × 𝐶_𝑛 have?
   2. Show that 𝐶_𝑛 × 𝐶_𝑛 is 4-regular. One way is to let (𝑝, 𝑞) be any vertex of 𝐶_𝑛 × 𝐶_𝑛 and explicitly show that (𝑝, 𝑞) is adjacent to exactly four vertices.
   3. How many edges does 𝐶_𝑛 × 𝐶_𝑛 have?

2. We discuss Cartesian products of regular graphs more generally.
   1. Let 𝐺 be an 𝑎-regular graph and 𝐻 be a 𝑏-regular graph. Show that 𝐺 × 𝐻 is an (𝑎 + 𝑏)-regular graph.
   2. (⋆) Let 𝐺 be an 𝑎-regular graph.  Show that 𝐺^𝑘 (this is the 𝑘-fold cartesian product 𝐺 × 𝐺 × ⋯ × 𝐺) is a 𝑘𝑎-regular graph.  If you use mathematical induction, you may assume that 𝐺^𝑘 is isomorphic to 𝐺 × 𝐺𝑘−1.

3. Let 𝑄₃ be the 3-cube with vertex set

𝑉 = {000,001,010,011,100,101,110,111}.

1. Give an example of two vertices such that if we delete them both, the graph 𝐺 that remains is a 6-cycle.
2. Suppose we start with 𝑄₃ and produce the graph 𝐻 by identifying vertices 000 and 011; we call the resulting vertex 𝑤. What is the degree of 𝑤 in 𝐻?
3. (⋆) What is the smallest number of edges we could delete from 𝑄₃ so that the resulting graph is disconnected?  Justify your answer.

4. Let 𝐺 be a graph with at least one edge.
   1. Show that for any 𝑘 ≥ 1, if 𝑊 is a walk in 𝐺 of length 𝑘, then there exists a walk 𝑊^′ in 𝐺 of length 𝑘 + 1. This shows that 𝐺 cannot have a longest walk.
   2. Show that there is some upper bound 𝑏 on the length of a path in 𝐺. This means that if 𝑊 is a walk of length 𝑘 > 𝑏, then a vertex must be repeated in 𝑊.  This shows that 𝐺 must have a

longest path.