CS 453 Graph Theory Assignment 3 solution

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1. Let 𝑛 β‰₯ 3 be given and let 𝐢𝑛 be the 𝑛-cycle whose vertices are
{0,1,2, … , 𝑛 βˆ’ 1}. Recall that π‘Žπ‘ is an edge if and only if π‘Ž = 𝑏 + 1 mod
𝑛 or 𝑏 = π‘Ž + 1 mod 𝑛.
a. How many vertices does 𝐢𝑛 Γ— 𝐢𝑛 have?
b. Show that 𝐢𝑛 Γ— 𝐢𝑛 is 4-regular. One way is to let (𝑝, π‘ž) be any vertex
of 𝐢𝑛 Γ— 𝐢𝑛 and explicitly show that (𝑝, π‘ž) is adjacent to exactly four
vertices.
c. How many edges does 𝐢𝑛 Γ— 𝐢𝑛 have?
2. We discuss Cartesian products of regular graphs more generally.
a. Let 𝐺 be an π‘Ž-regular graph and 𝐻 be a 𝑏-regular graph. Show that
𝐺 Γ— 𝐻 is an (π‘Ž + 𝑏)-regular graph.
b. (⋆) Let 𝐺 be an π‘Ž-regular graph. Show that 𝐺
π‘˜
(this is the π‘˜-fold
cartesian product 𝐺 Γ— 𝐺 Γ— β‹― Γ— 𝐺) is a π‘˜π‘Ž-regular graph. If you use
mathematical induction, you may assume that 𝐺
π‘˜
is isomorphic to
𝐺 Γ— 𝐺
π‘˜βˆ’1
.
3. Let 𝑄3 be the 3-cube with vertex set
𝑉 = {000,001,010,011,100,101,110,111}.
a. Give an example of two vertices such that if we delete them
both, the graph 𝐺 that remains is a 6-cycle.
b. Suppose we start with 𝑄3 and produce the graph 𝐻 by
identifying vertices 000 and 011; we call the resulting vertex 𝑀.
What is the degree of 𝑀 in 𝐻?
c. (⋆) 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.
a. Show that for any π‘˜ β‰₯ 1, if π‘Š is a walk in 𝐺 of length π‘˜, then
there exists a walk π‘Šβ€²
in 𝐺 of length π‘˜ + 1. This shows that 𝐺
cannot have a longest walk.
b. Show that there is some upper bound 𝑏 on the length of a path
in 𝐺. This means that if π‘Š is a walk of length π‘˜ > 𝑏, then a
vertex must be repeated in π‘Š. This shows that 𝐺 must have a
longest path.