Graph Theory MATH 3116
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This 5 page Class Notes was uploaded by Mrs. Dangelo Fahey on Sunday October 25, 2015. The Class Notes belongs to MATH 3116 at University of North Carolina - Charlotte taught by Gabor Hetyei in Fall. Since its upload, it has received 78 views. For similar materials see /class/228902/math-3116-university-of-north-carolina-charlotte in Mathematics (M) at University of North Carolina - Charlotte.
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Date Created: 10/25/15
Sample Test 1 1 Are the two graphs shown in the picture isomorphic Iustify your claim G1 has 11 edges while G2 has 12 edges gt G1 and G2 are NOT isomorphic 2 State and prove the degree sum formula Sum of degrees 2 times the edges in G 2 do 2eG v E G Proof Summing degrees counts each edge twice since each edge has 2 endpoints and each edge connects 2 vertices W Explain how the degreesum formula may be used to prove that a mountain climber puzzlequot always has a solution If starting positions are vertices x1 and X2 then they have an odd degree Based on the sum of degree theorem one other vertex with an odd degree exists In the graph the only other odd degree vector is the peak 2 so a path from x1 x2 to z Z w 4 What is the number of edges in the ndimensional hypercube Edges in n hypercube 271 1 n 5 What is the number of edges in the complete bipartite graph Km Complete bipartite graph every vertex of the rst set is connected to every vertex of the second set Km gt vertices m n Km gt edges m n 6 Give a necessary and suf cient condition for a graph to be bipartite and outline the proof of your claim Bipartite graphs do not contain any oddlength cycles and the vertices can be split into 2 disjoint sets Proof Let G v e and v A U B such that A n B at 0 and that all edges 8 E E are such that e is of the form 1 b where a E E and b E E 7 State and outline the proof of Euler39s formula for connected planar graphs r ev2 0 H 7 evl 11 34453th l 0 44quot to39cjjl S 393 Alva e719 5 1 2 AJ1 152 e 3 4quot A connected planar graph has 6 vertices and 3 regions What is the number ofits edges G6e gtre v2 gterv 27edges Prove that in a planar graph the number 9 of edges does not exceed 3v 6 where v is the number of vertices Planar graph sum of faces degrees 2 times the edges each region in plane graph m have I 2 3 3rZe gtr S Ee ua v2 S e gt v2 20 gte S3v 6 Use the result in the previous question to show that K5 is not planar e S3v 6 gt10 S35 6 gt10 9 gtNotPlanar Use the circlechord method to show that K33 is not planar Step 1 Find llamilton graph Step 2 Draw llamilton as circle L3 quotgt 39 L 39gt Ll 12 State the 4color theorem and explain how it applies to map coloring In particular does it mean that every map can be colored using four colors 4color theorem any map in plane can be colored using 4 colors in such a way that regions sharing a common boundary do not share the same color Yes every map can be colored using 4 colors 13 At least how many times do you need to lift your hand to draw the graph shown below Iustify your answer H l If more than 2 odd vertices in graph then more than 1 stroke is necessary Since the graph has 4 odd vertices which is greater than 2 more than 1 stroke will be needed Give a necessary and suf cient condition for the existence of an Eulerian cycle in a graph Outline the proof All vertices must have an even degree For which values of n does the complete graph Kquot have an Eulerian cycle For odd values of n each vertex of Kn has an even degree The maximum degree in a graph is d What can you say about its edge chromatic number Edge chromatic number fewest number of colors needed to color each edge of G e d or d 1 Based on Vizing39s theorem Grinberg39s theorem states that a planar graph that has a Hamilton circuit satis es the formula 2039 2w r5 0 Here riresp no is the number of regions that have i sides and lay inside resp outside the Hamilton circuit Use Grinberg39s theorem to show that the graph shown below has no Hamiltonian circuit 3regionsgtr3 13 1 r4 r4 1 r7 13 1 2039 2n T5 0 103 Ts 204 T4 507 T7 gt i 1 i 2 i 5 at 0 No Hamiltonian Circuit 18 Now use the three simple rules about building a Hamilton circuit to show that the graph in the previous question has no Hamilton circuit Rule 1 If a vertex X has degree 2 both of the edges incident to X must be part of any Hamilton circuit Rule 2 No proper sub circuit that is a circuit not containing all vertices can be formed when building a Hamilton circuit Rule 3 Once the Hamilton circuit is required to use two edges at a vertex X all other unused edges incident at X must be removed from consideration No Hamilton circuit because of sub circuits present 19 What is the chromatic number of a wheel graph with n vertices Based on parity of n vertices chromatic number 3 or 4 Ifn is even then 4 colors Ifn is odd then 3 colors 20 What is the chromatic polynomial of a path of length n Prove your formula 1 r L 3 1 L MS kl 1153 k kb 2 1 Prove that the chromatic polynomial PkG of a graph G satis es the recurrence e Here e is an edge of G and G e respectively Ge are the graphs obtained by deleting respectively contracting the edge e EXplain how the above recurrence may be used to prove that PkG is a polynomial What is the basis for the induction 31631 3 39 7 77 7 7 7 7 7 pm 7 7 7 EJ743157 7 7 77 Pp llebn 7 12377 7 715 7 g hwy 34747173371245 7 i 77 77L7IkJY LCJgiQbeZ37 i kh h f k 777 77777 V7 77 HkDM 3qu Pewb 46b l w p 4u lge1291 i quotrug 6g gamim 7 L ng M kL Zk 1amp3 aumgi qw ELL DLL5LY 77 7A
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