1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchang- ing the two partite sets) if and only if it is connected.
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- 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.4. [10 marks] Find both a matching of maximum size and a vertex cover of minimum size in the following bipartite graph. Prove that your answer is correct. ย ພI 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.
- 1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove: if all vertices have degree either 1 or at least 4, then T has at least 2(n + 1)/3 leaves. (4 pt)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)?The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n > 3, and connect this new vertex to each of n vertices by new edges. Match between each statement (a)- (d)) and a graph ((1)-(5)) such that the chosen graph satisfies the statement. (1) C, (2) C10 (3) W, (4) W, (5) W 10 A graph with the sum of degrees is 28. Choose... A simple and bipartite graph Choose... A simple graph with Hamiltonian circuit and vertex of degree 10. Choose... : A regular graph and not bipartite graph Choose...
- 1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 2317. Answer these two questions:(a) Find all the nonisomorphic complete bipartite graphs G = (V, E), where |V | = 6.(b) How many nonisomorphic complete bipartite graphs G = (V, E), satisfy |V | = n ≥ 2?Solve this Discrete Math problem.
- Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits of length three, then e ≤ 2v − 4. (Show work)Solve 5b1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.