Instructions: 1. Give geometric interpretation and graphs where required. 2. Give your original work. 3. Use the recommended references and books. Abbott H.L., Lower bounds for some Ramsey numbers. Discr. Math. 2 (1972), 289-293. [393] Abeledo H. and G. Isaak, A characterization of graphs that ensure the existence of a stable matching. Math. Soc. Sci. 22 (1991), 93-96. [136] Aberth O., On the sum of graphs. Rev. Fr. Rech. Opér. 33 (1964), 353-358. [194] Acharya B.D. and M. Las Vergnas, Hypergraphs with cyclomatic number zero, triangulated graphs, and an inequality. J. Comb. Th. B 33 (1982), 52–56. [327] Ahuja R.K., T.L. Magnanti, and J.B. Orlin, Network Flows. Prentice Hall (1993). [97, 145, 176, 180, 185, 190) Aigner M., Combinatorial Theory. Springer-Verlag (1979). [355, 360, 373] Aigner M., Graphentheorie. Eine Entwicklung aus dem 4-Farben Problem. B.G. Teubner Verlagsgesellschaft (1984) (English transl. BCS Assoc., 1987). [258] Ajtai M., V. Chvátal, M.M. Newborn and E. Szemerédi, Crossing-free subgraphs. Theory and practice of combinatorics, Ann. Discr. Math. 12 (1982), 9-12. [264] Ajtai M., J. Komlós, and E. Szemerédi, A note on Ramsey numbers. J. Comb. Th (A) 29 (1980), 354-360. [51, 385] Ajtai M., J. Komlós, and E. Szemerédi, Sorting in clog parallel steps. Combi- natorica 3 (1983), 1-19. [463] Akiyama J., H. Era, S.V. Gervacio and M. Watanabe, Path chromatic numbers of graphs. J. Graph Th. 13 (1989), 569-575. [271] Akiyama J, and F. Harary, A graph and its complement with specified properties, IV: Counting self-complementary blocks. J. Graph Th. 5 (1981), 103-107. [32] Albertson M.O. and E.H. Moore, Extending graph colorings. J. Comb. Th. (B) 77 (1999), 83-95. [204] Alekseev V.B. and V.S. Gončakov, The thickness of an arbitrary complete graph (Russian). Mat. Sb. (N.S.) 101(143) (1976), 212-230. [271] No AI, AI means Downvote. Problem 6: Knot Theory and Graphs on 3-Manifolds Exploring graphs within three-dimensional manifolds opens the door to connections with knot theory. 1. Graph Embeddings and Knot Complements: Show that for any knot K in R3, there exists a graph G whose embedding in R³ has a fundamental group isomorphic to the fundamental group of the knot complement R3- K. • For a given graph embedding, describe how to compute linking numbers for each pair of cycles and relate this to knot invariants. 2. Torus Knots and Graph Lattices: Given a torus knot T(p,q), embed a corresponding lattice graph on the surface of the torus that "wraps" in p and q directions. Prove that the fundamental group of this lattice graph embedding reflects the structure of the torus knot's group. • Analyze how changes in p and q values affect the degree and chromatic properties of the graph on the torus. 3. 3-Manifold Invariants from Graph Embeddings: • For a graph G embedded in a 3-manifold M, calculate the Heegaard genus of M in terms of cycles within G. Show how this genus changes if G undergoes an operation such as edge contraction.

Algebra & Trigonometry with Analytic Geometry
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Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 29E
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Instructions:
1. Give geometric interpretation and graphs where required.
2. Give your original work.
3. Use the recommended references and books.
Abbott H.L., Lower bounds for some Ramsey numbers. Discr. Math. 2 (1972),
289-293.
[393]
Abeledo H. and G. Isaak, A characterization of graphs that ensure the existence
of a stable matching. Math. Soc. Sci. 22 (1991), 93-96.
[136]
Aberth O., On the sum of graphs. Rev. Fr. Rech. Opér. 33 (1964), 353-358. [194]
Acharya B.D. and M. Las Vergnas, Hypergraphs with cyclomatic number zero,
triangulated graphs, and an inequality. J. Comb. Th. B 33 (1982), 52–56. [327]
Ahuja R.K., T.L. Magnanti, and J.B. Orlin, Network Flows. Prentice Hall (1993).
[97, 145, 176, 180, 185, 190)
Aigner M., Combinatorial Theory. Springer-Verlag (1979). [355, 360, 373]
Aigner M., Graphentheorie. Eine Entwicklung aus dem 4-Farben Problem. B.G.
Teubner Verlagsgesellschaft (1984) (English transl. BCS Assoc., 1987). [258]
Ajtai M., V. Chvátal, M.M. Newborn and E. Szemerédi, Crossing-free subgraphs.
Theory and practice of combinatorics, Ann. Discr. Math. 12 (1982), 9-12. [264]
Ajtai M., J. Komlós, and E. Szemerédi, A note on Ramsey numbers. J. Comb. Th
(A) 29 (1980), 354-360.
[51, 385]
Ajtai M., J. Komlós, and E. Szemerédi, Sorting in clog parallel steps. Combi-
natorica 3 (1983), 1-19.
[463]
Akiyama J., H. Era, S.V. Gervacio and M. Watanabe, Path chromatic numbers
of graphs. J. Graph Th. 13 (1989), 569-575.
[271]
Akiyama J, and F. Harary, A graph and its complement with specified properties,
IV: Counting self-complementary blocks. J. Graph Th. 5 (1981), 103-107. [32]
Albertson M.O. and E.H. Moore, Extending graph colorings. J. Comb. Th. (B) 77
(1999), 83-95.
[204]
Alekseev V.B. and V.S. Gončakov, The thickness of an arbitrary complete graph
(Russian). Mat. Sb. (N.S.) 101(143) (1976), 212-230.
[271]
No AI, AI means Downvote.
Problem 6: Knot Theory and Graphs on 3-Manifolds
Exploring graphs within three-dimensional manifolds opens the door to connections with knot
theory.
1. Graph Embeddings and Knot Complements:
Show that for any knot K in R3, there exists a graph G whose embedding in R³ has a
fundamental group isomorphic to the fundamental group of the knot complement R3-
K.
• For a given graph embedding, describe how to compute linking numbers for each pair of
cycles and relate this to knot invariants.
2. Torus Knots and Graph Lattices:
Given a torus knot T(p,q), embed a corresponding lattice graph on the surface of the
torus that "wraps" in p and q directions. Prove that the fundamental group of this lattice
graph embedding reflects the structure of the torus knot's group.
• Analyze how changes in p and q values affect the degree and chromatic properties of the
graph on the torus.
3. 3-Manifold Invariants from Graph Embeddings:
• For a graph G embedded in a 3-manifold M, calculate the Heegaard genus of M in terms
of cycles within G. Show how this genus changes if G undergoes an operation such as
edge contraction.
Transcribed Image Text:Instructions: 1. Give geometric interpretation and graphs where required. 2. Give your original work. 3. Use the recommended references and books. Abbott H.L., Lower bounds for some Ramsey numbers. Discr. Math. 2 (1972), 289-293. [393] Abeledo H. and G. Isaak, A characterization of graphs that ensure the existence of a stable matching. Math. Soc. Sci. 22 (1991), 93-96. [136] Aberth O., On the sum of graphs. Rev. Fr. Rech. Opér. 33 (1964), 353-358. [194] Acharya B.D. and M. Las Vergnas, Hypergraphs with cyclomatic number zero, triangulated graphs, and an inequality. J. Comb. Th. B 33 (1982), 52–56. [327] Ahuja R.K., T.L. Magnanti, and J.B. Orlin, Network Flows. Prentice Hall (1993). [97, 145, 176, 180, 185, 190) Aigner M., Combinatorial Theory. Springer-Verlag (1979). [355, 360, 373] Aigner M., Graphentheorie. Eine Entwicklung aus dem 4-Farben Problem. B.G. Teubner Verlagsgesellschaft (1984) (English transl. BCS Assoc., 1987). [258] Ajtai M., V. Chvátal, M.M. Newborn and E. Szemerédi, Crossing-free subgraphs. Theory and practice of combinatorics, Ann. Discr. Math. 12 (1982), 9-12. [264] Ajtai M., J. Komlós, and E. Szemerédi, A note on Ramsey numbers. J. Comb. Th (A) 29 (1980), 354-360. [51, 385] Ajtai M., J. Komlós, and E. Szemerédi, Sorting in clog parallel steps. Combi- natorica 3 (1983), 1-19. [463] Akiyama J., H. Era, S.V. Gervacio and M. Watanabe, Path chromatic numbers of graphs. J. Graph Th. 13 (1989), 569-575. [271] Akiyama J, and F. Harary, A graph and its complement with specified properties, IV: Counting self-complementary blocks. J. Graph Th. 5 (1981), 103-107. [32] Albertson M.O. and E.H. Moore, Extending graph colorings. J. Comb. Th. (B) 77 (1999), 83-95. [204] Alekseev V.B. and V.S. Gončakov, The thickness of an arbitrary complete graph (Russian). Mat. Sb. (N.S.) 101(143) (1976), 212-230. [271] No AI, AI means Downvote. Problem 6: Knot Theory and Graphs on 3-Manifolds Exploring graphs within three-dimensional manifolds opens the door to connections with knot theory. 1. Graph Embeddings and Knot Complements: Show that for any knot K in R3, there exists a graph G whose embedding in R³ has a fundamental group isomorphic to the fundamental group of the knot complement R3- K. • For a given graph embedding, describe how to compute linking numbers for each pair of cycles and relate this to knot invariants. 2. Torus Knots and Graph Lattices: Given a torus knot T(p,q), embed a corresponding lattice graph on the surface of the torus that "wraps" in p and q directions. Prove that the fundamental group of this lattice graph embedding reflects the structure of the torus knot's group. • Analyze how changes in p and q values affect the degree and chromatic properties of the graph on the torus. 3. 3-Manifold Invariants from Graph Embeddings: • For a graph G embedded in a 3-manifold M, calculate the Heegaard genus of M in terms of cycles within G. Show how this genus changes if G undergoes an operation such as edge contraction.
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