Student name Math 104 Graph Theory by ztn18049

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Student name
Math 104 Graph Theory
Homework 11
4/30/2009 (Thursday)
Introduction to Graph Theory, West
Section 7.2 7, 10, 14, 26
Section 8.3 17
Section 1.4 26
H AMILTONIAN GRAPHS

7.2.7 A mouse eats its way through a 3 × 3 × 3 cube of cheese by eating all the 1 × 1 × 1
subcubes. If it starts at a corner subcube and always moves on to an adjacent subcube (sharing a
face of area 1), can it do this and eat the center subcube last? Give a method or prove impossible.
(Ignore gravity.)

7.2.10 Hamiltonian vs. Eulerian.
(a) Find a 2-connected non-Eulerian graph whose line graph is Hamiltonian.
(b) Prove that L(G) is Hamiltonian if and only if G has a closed trail that contains at least one
endpoint of each edge.

7.2.14 A graph G is uniquely k-edge-colorable if all proper k-edge-colorings of G induce the
same partition of the edges. Prove that every uniquely 3-edge-colorable 3-regular graph is Hamil-
tonian.

7.2.26 Prove that if G fails Chv´ tal’s condition, then G has at least n − 2 edges. Conclude from
a
this that the maximum number of edges in a simple non-Hamiltonian n-vertex graph is n−1 + 1.
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R AMSEY THEORY

8.3.17 Ramsey numbers for r = 2 and multiple colors.
(a) Let p = (p1 , . . . , pk ) and let qi be obtained from p by subtracting 1 from pi but leaving the
other coordinates unchanged. Prove that R(p) ≤ k R(qi ) − k + 2.
i=1
(b) Prove that R(p1 + 1, . . . , pk + 1) ≤ (p1 +···+p!k )! .
p1 !···pk

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E ULERIAN GRAPHS AND DIGRAPHS

1.4.26 Arrange seven 0’s and seven 1’s cyclically so that the 14 strings of four consecutive bits
are all the 4-digit binary strings other than 0101 and 1010.

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