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- Let T be a tree of order n and suppose that all vertices of T have degree 1 or degree 3. Prove that T contains exactly n-2/2 vertices of degree 3What is the chromatic number of a full m-ary tree of height h?A (general) bipartite graph G is a simple graph whose vertex set can be partitioned into two disjoint nonempty subsets V1 and V2 such that vertices in V1 may be connected to vertices in V2 , but no vertices in V1 are connected to other vertices in V1 and no vertices in V2 are connected to other vertices in V2 . For example, the bipartite graph G illustratedin (i) can be redrawn as shown in (ii). From the drawing in (ii), you can see that G is bipartite with mutually disjoint vertex sets V1 = { v1 , v3 , v5 } and V2 = { v2 , v4 , v6 } .Find which of the following graphs are bipartite. Redraw the bipartite graphs so that their bipartite nature is evident
- The Feedback Vertex Set problem is defined as follows:Given: a graph G and a positive integer kQuestion: does G have a set S of at most k vertices such that the subgraph inducedby the complement of S is acyclic?In other words, the problem asks whether deleting at most k vertices from a graphmakes it acyclic.Assume you found a trusted polynomial-time algorithm, A, for the decision versionof the problem. Write a polynomial-time algorithm that uses A to solve the searchversion.2. Answer the same question for the Graph Coloring problem.Prove that for every odd n ≥ 5 there exist a graph of n+1 vertices n of which have degree 3, and the last one has degree n not equal 3Prove that every connected planar graph with less than 12 vertices has a vertex of degree at most 4. [Hint: Assume that every vertex has degree at least 5 to obtain a lower bound on e (together with the upper bound on e in the corollary) that implies v ≥ 12.]
- i. Draw a connected bipartite graph with 5 labelled vertices, {a, b, c, d, e} = V and 5 edges.Based on the graph you’ve drawn, give the corresponding partition π = {V1, V2} and the relationρ ⊂ V1 × V2 corresponding with the edges.ii. Let A be a set of six elements and σ an equivalence relation on A such that the resultingpartition is {{a, b, f }, {c, e}, {d}}. Draw the directed graph corresponding with σ on A.iii. Draw a directed graph with 5 vertices and 10 edges (without duplicating any edges) repre-senting a relation ρ that is reflexive and antisymmetric, but not symmetric or transitive. Notehow these properties can be identified from the graph.If G is a finite graph, the independence number α(G) is the maximumnumber of pairwise nonadjacent vertices of G. The chromatic number χ(G)of G is the minimum number of colors in a coloring of the vertices of Gwith the property that no two adjacent vertices have the same color. Provethat in any graph G with n vertices, n ≤ α(G)χ(G)