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. 2 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 1 2 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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