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Name ________________________________
Math 575
Exam #1B
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1. (a). Explain the difference between a plane graph and a planar graph.
Solution: A planar graph is one that can be drawn in the plane so that no two
edges cross; A plane graph refers to an actual drawing of a planar graph without
crossings.
(b). Define: radius of a graph.
Solution: The radius of a graph is min{e(v) : v !V (G)} where e(v) is the
eccentricity of v.
(c). Define: H is a spanning subgraph of the graph G if
Solution: H is a spanning subgraph of G if V (H ) = V (G) and E(H ) ! E(G) .
2.
(a). Find the Prüffer Code of the tree below.
Solution: ( 7,!1,!2,!1,!7,!2,!8 )
(b). Find the tree whose Prüffer Code is (4, 2, 2, 7, 1, 7, 2).
Solution:
Name ________________________________
3. How many spanning trees are there for the graph below?
Solution:
Consider cases where and the answer is (12 + 9 ! 3 + 5 ! 7 + 5 ! 4 ! 3) ! 4 = 536 .
4. (a). If the degree sequence of G is 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 5, then determine
the degree sequence of the complement of G.
Solution: G has 11 vertices and so deg G (v) = 10 ! deg G (v) .
Hence the degree sequence is 5, 5, 6, 7, 7, 7, 7, 8, 8, 9, 9.
(b). Suppose that G is a graph that is regular of degree 6.
Complete the following argument that G cannot contain a bridge.
Proof. Suppose that e = ab is a bridge of G. Then G – e contains exactly two
components A and B with a in A and b in B.
But then in the subgraph induced by A, the vertex a has degree 5 and the
other vertices of A have degree 6. Thus this graph would have an odd number
of vertices of odd degree and that is impossible.
5. (a). How many distinct trees on the vertex set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are
there in which deg(1) = 4 and deg(3) = 2 ?
! 8$
Solution: # & ' 5 ' 8 4 .
" 3%
Count the number of Prüffer codes. Choose 3 of the 8 slots for the 1’s and then choose
one of the 5 remaining slots for the 3 and finally place the remaining 8 labels into the
remaining 4 slots.
(b). How many trees are there on {1, 2, 3, …, 13} that have exactly four end-vertices
and exactly one vertex of degree 4?
! 13$
! 11$
! 13$ ! 11$
Solution: # & ' 9 ' # & ' 8! = # & # & ' 9!
" 4%
" 3%
" 4 %" 3%
First choose the 4 end-vertices. Then from the remaining 9 numbers, choose the vertex
of degree 4. Then choose 3 of the 11 available slots for the chosen vertex of degree 4
and then place the remaining 8 numbers into the remaining 8 slots with each number
used exactly once.
Name ________________________________
6. Let G be any tree on 8 vertices. Explain how you know that G is not planar.
Hint: Think number of edges. Do not use Kuratowski’s Theorem.
! 8$
Solution: G has 7 edges and so G has # & ' 7 = 28 ' 7 = 21 edges.
" 2%
However the maximum number of edges that a planar graph on 8 vertices can
have is 3( 8 ) ! 6 = 18 . So, since 21 > 18, G is not planar.
(b). Show that the graph below is not planar by applying Kuratowski’s Theorem.
Solution: The graph below represents a subgraph that is a subdivision of K 3, 3 .
7. Let G be a plane graph with 21 faces. Three of the faces are bounded by 4 edges
and eighteen of the faces are bounded by 3-edges. All the vertices of G have degree
4 or 5. Suppose that G has a vertices of degree 4 and b vertices of degree 5.
Find the values of a and b.
Solution: We have 2m =
$
!F = 3 % 4 + 18 % 3 = 66 & m = 33 .
F "#(G )
(Here !(G) represents the set of all faces of G.)
From Euler’s Theorem, n ! m + f = 2 " n = m ! f + 2 = 33 ! 21 + 2 = 14 .
4a + 5b = 2m = 66,!!!a + b = n = 14 .
Solving these simultaneously gives a = 4 and b = 10 .
Name ________________________________
8. (a). The two connected graphs G and H below are isomorphic.
Complete the isomorphism below from G to H.
a ! 8,!!b ! 7,!!c !!6!,!!d ! __,!!e ! ___,!! f ! __,!!g ! __,!h ! __
Solution: a ! 8,!!b ! 7,!!c !!6!,!!d ! 5,!!e ! 4,!! f ! 3,!!g ! 2,!h ! 1 .
(b). Suppose that G is a graph with 14 vertices such that for every vertex v,
G – v has 48 edges. Then G is regular of degree 8
(Show your work to justify your answer.)
Solution: The number of edges of G is m =
# m (G ! v )
v"V (G )
Thus the degree of each vertex is 56 ! 48 = 8 .
n!2
=
14 $ 48
= 56 .
14 ! 2
9. The picture below shows an unfinished embedding of K7 in a torus. You don’t
have to finish it. Just show how to add the edge between vertices x and c and x
and f.
Solution:
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10. Prove that if G is a planar graph, then the vertices of G can be colored using at
most five different colors.
Solution: See your notes.