1.2.10. (-) Prove or disprove: a) Every Eulerian bipartite graph has an even number of edges. b) Every Eulerian simple graph with an even number of vertices has an even num- ber of edges.
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- I want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.(Note: Please provide a precise answer and explain briefly which is not provided on Chegg or Bartleby.)2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m, n) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W , there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of W to any other vertex of W. Use ways to select the edges to show that this graph has m.n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2 (NOTE: Please elaborate on the answver and explain. Please do not copy-paste the answer from the internet or from Chegg.)
- 3. (a) Is it possible to have a 4-regular graph with 15 vertices? If no, explain why. If yes, construct such a graph. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers r, the graph G contains exactly r vertices of degree r, prove that two-thirds of the vertices of G have odd degree.2.12 Prove that a 3-regular graph has a cut vertex if, and only if, it has some bridge.(b) Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are degree 1, and 3 of its vertices are degree 2. Prove that G is planar. (Hint: Kuratowski) (c) Let G be any simple connected planar graph with n vertices and e edges. Suppose there are exactly y vertices of degree 2. Assume that n - y > 3. Prove that e < 3n - y - 6. (Hint: Explain why the degree-2 vertices can be erased, and how to take care of any resulting loops or multiple edges.) (d) Suppose that a connected simple graph G' has exactly 10 vertices of degree 4, 8 vertices of degree 5, and all other vertices have degree 7. Find the maximum possible number of degree-7 vertices G could have, so that G would still be planar.
- Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?Which one is correct?
- Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has degree greater than or equal to "1, then the diameter of G is 2 or less. If G is a (not necessarily simple) graph with n vertices n-1 2 where each vertex has degree greater than or equal to ", is the diameter of G necessarily 2 or less? Either prove that the answer to this question is "yes" or give a counterexample.can I please have the answer for 19

