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 n 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. Properti Problem 2: Geometric Graph Properties on Non-Orientable Surfaces Problem 3: Homology and Fundamental Groups of Graphs on Surfaces Homology and fundamental groups provide powerful tools for studying graph embeddings and surface properties. 1. Fundamental Group Calculation: • Consider a graph G embedded on a torus T2. Calculate the fundamental group T1 (G) of the embedded graph, assuming G is a cycle graph C, that wraps around the torus. Extend this calculation to an n-cycle on a genus-2 surface. 2. Homology of Graph Complexes: For a given graph G embedded on a surface S, (genus g), calculate the first homology group H₁(S) in terms of cycles and loops within G. Prove that the rank of H₁ (Sg) is 2g+ |V|-|E| for any connected graph G embedded on a surface S. 3. Mapping Class Group Action: Show that the mapping class group of a surface S, acts on the fundamental group of a graph embedded on it. Specifically, consider how the automorphisms of the fundamental group change under a twist in the embedding.

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 n 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. Properti Problem 2: Geometric Graph Properties on Non-Orientable Surfaces Problem 3: Homology and Fundamental Groups of Graphs on Surfaces Homology and fundamental groups provide powerful tools for studying graph embeddings and surface properties. 1. Fundamental Group Calculation: • Consider a graph G embedded on a torus T2. Calculate the fundamental group T1 (G) of the embedded graph, assuming G is a cycle graph C, that wraps around the torus. Extend this calculation to an n-cycle on a genus-2 surface. 2. Homology of Graph Complexes: For a given graph G embedded on a surface S, (genus g), calculate the first homology group H₁(S) in terms of cycles and loops within G. Prove that the rank of H₁ (Sg) is 2g+ |V|-|E| for any connected graph G embedded on a surface S. 3. Mapping Class Group Action: Show that the mapping class group of a surface S, acts on the fundamental group of a graph embedded on it. Specifically, consider how the automorphisms of the fundamental group change under a twist in the embedding.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 29E

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