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] 3. Mapping Class Group Action: No AI, AI means Downvote. ⚫ 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. 4. Covering Spaces and Lifts: of ⚫ Describe the process of lifting a graph embedding from a torus T2 to its universal cover, the plane R². Discuss how the covering space theory can be applied to determine if the embedding is unique and analyze the types of symmetry in the lifted graph. nd 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 Sg.

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] 3. Mapping Class Group Action: No AI, AI means Downvote. ⚫ 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. 4. Covering Spaces and Lifts: of ⚫ Describe the process of lifting a graph embedding from a torus T2 to its universal cover, the plane R². Discuss how the covering space theory can be applied to determine if the embedding is unique and analyze the types of symmetry in the lifted graph. nd 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 Sg.

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

See similar textbooks