MATH 4707 PROBLEM SET 6 1. Required problems • Exercises

MATH 4707 PROBLEM SET 6
FINAL VERSION; DUE WEDNESDAY 4/29
1. Required problems
• Exercises from the text: 13.4.2, 13.4.4, 13.4.5, 13.4.8
(1) Show that it is possible to draw K5 on a torus. What are the values of V , E, F for this
drawing? [Note: it may be helpful to think of a torus as a square with the opposite edges
identified, as in some old video games, so that moving off the left edge returns you on the
right edge similarly for the top and bottom. If you draw things like this then some edges
and faces will “wrap around.”]
(2) In this problem, let G = Kn,n be the complete bipartite graph with n vertices on each side.
(a) Show that CG (2) = 2.
(b) Show that CG (3) = 6 · 2n − 6.
(c) Make the following (very counter-intuitive!) conclusion: if n is sufficiently large (say
for example n = 10), the probability that randomly coloring G with three colors results
in a proper coloring is smaller than the probability that randomly coloring G with two
colors results in a proper coloring.
(3) (10 points) Recall that we defined a contracted graph as follows: given a graph G with vertices
a, b connected by an edge e = {a, b}, the contraction G/e has vertex set (V (G)r{a, b})∪{ab}
(i.e., all the vertices of G except a and b, with a new vertex added called “ab”) and edge
set {{c, d} : {c, d} ∈ E(G) and c, d 6= a, b} ∪ {{c, ab} : {c, a} ∈ E(G) or {c, b} ∈ E(G)}
(i.e., leave edges not touching a, b as they were, and connect old vertex c to the new vertex ab if and only if c was connected to at least one of a, b). For example, if G has vertices {w, x, y, z, c} and edges {{w, x}, {x, y}, {y, z}, {z, w}, {c, w}, {c, x}, {c, y}, {c, z}} (draw
it!) then the contraction G/{c, w} has the four vertices {cw, x, y, z} and the five edges
{{x, y}, {y, z}, {cw, x}, {cw, y}, {cw, z}}. Define T (G) to be the number of spanning trees
of a graph G.
(a) Suppose that G is a graph and {a, b} is an edge of G. Show that the number of spanning
trees of G that do not contain {a, b} is equal to T (G r {a, b}) (where G r {a, b} is the
result of deleting edge {a, b} from G).
(b) Suppose that G is a graph and {a, b} is an edge of G and no vertex is connected to both
a, b. Show that T (G/{a, b}) (this is a contraction) is the number of spanning trees of
G that do contain the edge {a, b}.
(c) Use the preceding parts to give a recursive formula for T (G) in terms of smaller graphs
(i.e., those with strictly fewer edges or with strictly fewer vertices) in the case that G
has an edge {a, b} such that a and b have no neighbors in common.
(d) Does the result of part (c) work without the hypothesis on edge {a, b}? (Hint: consider
K3 , and be careful about the definition of contraction: the result of contraction is always
a (simple) graph.) Can you explain how to fix it with multigraphs?
2. Optional problems
• Text: 11.3.1, 12.1.1, 12.3.6 (but please assume that the faces are regular pentagons and
hexagons, so that at each vertex we must have three faces meeting), 13.1.2, 13.3.2, 13.3.4,
13.4.3, 13.4.6
• Can K6 be drawn on a torus? K7 ?
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