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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

(a) Determine P(G, λ) for G = K1,3. b) For n e Z+, what is the chromatic polynomial for K1,n? What is its chromatic number?
(a) Consider the graph K2,3 shown in Fig. 11.91, and let Î» ˆˆ Z+ denote the number of colors available to properly color the vertices of K2,3. (i) How many proper colorings of K2,3 have vertices a, b colored the same? (ii) How many proper colorings of K2,3 have vertices a, b colored with
Find the chromatic number of the following graphs.(a) The complete bipartite graphs Km,n.(b) A cycle on n vertices, n ‰¥ 3.(c) The graphs in Figs. 11.59(d), 11.62(a), and 11.85.(d) The n-cube Qn, n ‰¥ 1.
If G is a loop-free undirected graph with at least one edge, prove that G is bipartite if an only if x (G) = 2.
(a) Determine the chromatic polynomials for the graphs in Fig. 11.92(b) Find x (G) for each graph.(c) If five colors are available, in how many ways can the vertices of each graph be properly colored?
Let G be a loop-free undirected graph on n vertices. If G has 56 edges and has 80 edges, what is n?
If G = (V, E) is an undirected graph, a subset D of V is called a dominating set if for all v ∈ V, either v ∈ D or v is adjacent to a vertex in D. If D is a dominating set and no proper subset of D has this property, then D is called minimal. The size of any smallest dominating set in G is
Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 11.94. For n ‰¥ 0, let an denote the number of ways one can select n of the edges in G so that no two edges share a common vertex. Find and solve a recurrence relation for an.
Consider the four comb graphs in parts (i), (ii), (iii), and (iv) of Fig. 11.96. These graphs have 1 tooth, 2 teeth, 3 teeth, and ft teeth, respectively. For n ‰¥ 1, let an count the number of independent subsets in {x1, x2, ... , xn, y1, y2, ..., yn}. Find and solve a recurrence relation
Consider the four graphs in parts (i), (ii), (iii), and (iv) of Fig. 11.97. If an counts the number of independent subsets of {x1, x2,....,xn, y1, y2, . . . . , yn}, where n ‰¥ 1, find and solve a recurrence relation for an.
For n ‰¥ 1, let the number of edges in Kn, and let a0 = 0. Find the generating function f(x) = ˆ‘ˆžn=0 anxn.
For the graph G in Fig. 11.98, answer the following questions.(a) What are Î³(G), Î²(G), and x(G)?(b) Does G have an Euler circuit or a Hamilton cycle?(c) Is G bipartite? Is it planar?
(a) Suppose that the complete bipartite graph Km,n contains 16 edges and satisfies m ≤ n. Determine ra, n so that Km,n possesses (i) An Euler circuit but not a Hamilton cycle; (ii) Both a Hamilton cycle and an Euler circuit. (b) Generalize the results of part (a).
If G = (V, E) is an undirected loop-free graph, the line graph of G, denoted L(G), is a graph with the set E as vertices, where we join two vertices e1, e2 in L(G) if and only if e1, e2 are adjacent edges in G.(a) Find L(G) for each of the graphs in Fig. 11.99.(b) Assuming that |V| = n and |E| = e,
Explain why each of the following polynomials in λ cannot be a chromatic polynomial. (a) λ4 - 5λ3 + 7λ2 - 6λ + 3 (b) 3λ3 - 4λ2 + λ (c) λ4 - 3λ3 + 5λ2 - 4λ
Determine the number of cycles of length 4 in the hypercube Qn.
(a) For all x, y e Z+, prove that x3y - xy3 is even. (b) Let V = {1,2,3,...,8, 9}. Construct the loop-free undirected graph G = (V, E) as follows: For m, n e V, m ≠ n, draw the edge [m, n] in G if 5 divides m + n or m - n. (c) Given any three distinct positive integers, prove that there are two
(a) For n ≥ 1, let Pn-1 denote the path made up of n vertices and n - 1 edges. Let an be the number of independent subsets of vertices in Pn-1. (The empty subset is considered one of these independent subsets.) Find and solve a recurrence relation for an. (b) Determine the number of independent
Suppose that G = (V, E) is a loop-free undirected graph. If G is 5-regular and |V| = 10, prove that G is nonplanar.
(a) If the edges of K6 are painted either red or blue, prove that there is a red triangle or a blue triangle that is a subgraph. (b) Prove that in any group of six people there must be three who are total strangers to one another or three who are mutual friends.
(a) Let G = (V, E) be a loop-free undirected graph. Recall that G is called self-complementary if G and are isomorphic. If G is self-complementary (i) determine |E| if |V| = n; (ii) prove that G is connected.(b) Let n ∈ Z+, where n = 4k (k e Z+) or n = 4k + 1 (k ∈ N). Prove that there exists a
(a) Show that the graphs G1 and G2, in Fig. 11.95, are isomorphic.(b) How many different isomorphisms f: G1 †’ G2 are possible here?
Are any of the planar graphs for the five Platonic solids bipartite?
(a) How many paths of length 5 are there in the complete bipartite graph K37? (Remember that a path such as v1 → v2 → v3 → v4 → v5 → v6 is considered to be the same as the path v6 → v5 → v4 → v3 → v2 → v1.) (b) How many paths of length 4 are there in K3,7? (c) Let m, n, p ∈ Z+
Let x = {1, 2, 3, ... , n}, where n ≥ 2. Construct the loop- free undirected graph G = (V, E) as follows: • (V): Each two-element subset of X determines a vertex of G. • (E): If v1, v2 ∈ V correspond to subsets {a, b} and {c, d}, respectively, of X, draw the edge {v1, v2] in G when {a, b}
If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a, b} of G either a or b is in K. The set K is a minimal covering if K - {x} fails to cover G for each x e K. The number of vertices in a smallest covering is called the covering number of G. (a)
(a) Draw the graphs of all nonisomorphic trees on six vertices. (b) How many isomers does hexane (C6H14) have?
Find two nonisomorphic spanning trees for the complete bipartite graph K2,3. How many nonisomorphic spanning trees are there for K2,3?
For n ∈ Z+, how many nonisomorphic spanning trees are there for K2,n?
For each graph in Fig. 12.7, determine how many nonidentical (though some may be isomorphic) spanning trees exist.
Let T = (V, E) be a tree where |V| = n. Suppose that for each v ∈ V, deg(v) = 1 or deg(v) ≥ m, where m is a fixed positive integer and m ≥ 2.(a) What is the smallest value possible for n?(b) Prove that T has at least m pendant vertices.
Let G = (V, E) be a loop-free connected undirected graph. Let H be a subgraph of G. The complement of H in G is the subgraph of G made up of those edges in G that are not in H (along with the vertices incident to these edges). (a) If T is a spanning tree of G, prove that the complement of T in G
Complete the proof of Theorem 12.5. The following statements are equivalent for a loop-free undirected graph G = (V, E). (a) G is connected, and |V| = |E| + 1. (b) G contains no cycles, and if a, b ∈ V with {a, b} ∉ E, then the graph obtained by adding edge {a, b} to G has precisely one cycle.
A labeled tree is one wherein the vertices are labeled. If the tree has n vertices, then {1, 2, 3,..., n} is used as the set of labels. We find that two trees that are isomorphic without labels may become nonisomorphic when labeled. In Fig. 12.8, the first two trees are isomorphic as labeled trees.
Let n ∈ Z+, n ≥ 3. If v is a vertex in Kn, how many of the nn-2 spanning trees of Kn have v as a pendant vertex?
Characterize the trees whose Prufer codes (a) Contain only one integer, or (b) Have distinct integers in all positions.
Show that the number of labeled trees with n vertices, k of which are pendant vertices, is (n - k)!S(n - 2, n - k) = (n!/k!)S(n - 2, n - k), where S(n - 2, n - k) is a Stirling number of the second kind. (This result was first established in 1959 by A. Renyi.)
Let G = (V, E) be the undirected graph in Fig. 12.9. Show that the edge set E can be partitioned as E1 ˆª E2 so that the subgraphs G1 = (V, E1), G2 = (V, E2) are isomorphic spanning trees of G.
(a) Let F1 = (V1, E1) be a forest of seven trees where |E1| = 40. What is |V1|? (b) If F2 = (V2, E2) is a forest with |V2| = 62 and |E2| = 51, how many trees determine F2?
(a) Verify that all trees are planar. (b) Derive Theorem 12.3 from part (a) and Euler's Theorem for planar graphs.
Give an example of an undirected graph G = (V, E) where |V| = |E| + 1 but G is not a tree.
(a) If a tree has four vertices of degree 2, one vertex of degree 3, two of degree 4, and one of degree 5, how many pendant vertices does it have? (b) If a tree T = (V, E) has v2 vertices of degree 2, v3 vertices of degree 3,..., and vm vertices of degree m, what are |V| and |E|?
If G = (V, E) is a loop-free undirected graph, prove that G is a tree if there is a unique path between any two vertices of G.
Answer the following questions for the tree shown in Fig. 12.28.(a) Which vertices are the leaves? (b) Which vertex is the root? (c) Which vertex is the parent of g? (d) Which vertices are the descendants of c? (e) Which vertices are the siblings of s? (f) What is the level number of vertex f? (g)
Prove Theorem 12.6 and Corollary 12.1. Theorem 12.6 Let T = (V, E) be a complete m-ary tree with |V| = n. If T has i leaves and i internal vertices, then (a) n = mi + 1; (b) ℓ = (m - 1)i + 1; and (c) i = (ℓ - 1)/(m - 1) = (n - 1)/m. Corollary 12.1. Let T be a balanced complete m-ary tree with i
With m, n, i, ℓ as in Theorem 12.6, prove that (a) n = (mℓ - 1 )/(m - 1). (b) ℓ = [(m - 1)n + 1/m.
(a) A complete ternary (or 3-ary) tree T = (V, E) has 34 internal vertices. How many edges does T have? How many leaves? (b) How many internal vertices does a complete 5-ary tree with 817 leaves have?
The complete binary tree T = (V, E) has V = {a, b, c, ...,i, j, k}. The post order listing of V yields d, e, b, h, i, f, j, k, g, c, a. From this information draw T if (a) The height of T is 3; (b) The height of the left subtree of T is 3.
For m ‰¥ 3, a complete m-ary tree can be transformed into a complete binary tree by applying the idea shown in Fig. 12.32.(a) Use this technique to transform the complete ternary decision tree shown in Fig. 12.27(b).(b) If T is a complete quaternary tree of height 3, what is the maximum
(a) At a men's singles tennis tournament, each of 25 players brings a can of tennis balls. When a match is played, one can of balls is opened and used, then kept by the loser. The winner takes the unopened can on to his next match. How many cans of tennis balls will be opened during this
Use a complete ternary decision tree to repeat Example 12.15 for a set of 12 coins, exactly one of which is heavier (and counterfeit).
Let T = (V, E) be a binary tree. In Fig. 12.29 we find the subtree of T rooted at vertex p. (The dashed line coming into vertex p indicates that there is more to the tree T than what appears in the figure.) If the level number for vertex u is 37,(a) What are the level numbers for vertices p, s, t,
Let T = (V, E) be a balanced complete m-ary tree of height h ≥ 2. If T has ℓ leaves and bh-1 internal vertices at level h - 1, explain why ℓ = mh-1 + (m - 1)bh-1.
Consider the complete binary trees on 31 vertices. (Here we distinguish left from right as in Example 12.9.) How many of these trees have 11 vertices in the left subtree of the root? How many have 21 vertices in the right subtree of the root?
Consider the following algorithm where the input is a rooted tree with root r.Step 1: Push r onto the (empty) stackStep 2: While the stack is not emptyPop the vertex at the top of the stack and record its labelPush the children - going from right to left - of this vertex onto the stack(The stack
Consider the following algorithm where the input is a rooted tree with root r.Step 1: Push r onto the (empty) stackStep 2: While the stack is not emptyIf the entry at the top of the stack is not markedThen mark it and push itschildren-right to left-onto the stackElsePop the vertex at the top of the
Let T = (V, E) be a rooted tree ordered by a universal address system, (a) If vertex v in T has address 2.1.3.6, what is the smallest number of siblings that v must have? (b) For the vertex v in part (a), find the address of its parent, (c) How many ancestors does the vertex v in part (a) have? (d)
For the tree shown in Fig. 12.30, list the vertices according to a preorder traversal, an inorder traversal, and a postorder traversal.
List the vertices in the tree shown in Fig. 12.31 when they are visited in a preorder traversal and in a postorder traversal.
(a) Find the depth-first spanning tree for the graph shown in Fig. 11.72(a) if the order of the vertices is given as (i) a, b, c, d, e, f, g, h; (ii) h, g, f, e, d, c, b, a; (iii) a, b, c, d, h, g, f, e. (b) Repeat part (a) for the graph shown in Fig. 11.85(i).
Find the breadth-first spanning trees for the graphs and prescribed orders given in Exercise 7. (a) Find the depth-first spanning tree for the graph shown in Fig. 11.72(a) if the order of the vertices is given as (i) a, b, c, d, e, f, g, h; (ii) h, g, f, e, d, c, b, a; (iii) a, b, c, d, h, g, f,
Let G = (V, E) be an undirected graph with adjacency matrix A(G) as shown here.Use a breadth-first search based on A(G) to determine whether G is connected.
(a) Give an example of two lists L1, L2, each of which is in ascending order and contains five elements, and where nine comparisons are needed to merge L1, L2 by the algorithm given in Lemma 12.1. (b) Let m, n e Z+ with m < n. Give an example of two lists L1, L2, each of which is in ascending
Apply the merge sort to each of the following lists. Draw the splitting and merging trees for each application of the procedure. (a) - 1, 0, 2, - 2, 3, 6, - 3, 5, 1, 4 (b) - 1, 7, 4, 11, 5, - 8, 15, - 3, - 2, 6, 10, 3
Related to the merge sort is a somewhat more efficient procedure called the quick sort. Here we start with a list L: a1, a2, . . . , an, and use a1 as a pivot to develop two sublists L1 and L2 as follows. For i > 1, if al < a1, place ax at the end of the first list being developed (this is L1 at
Prove that the function g used in the second method to analyze the (worst-case) time-complexity of the merge sort is monotone increasing.
Construct an optimal prefix code for the symbols a, b, c, . . . , i, j that occur (in a given sample) with respective frequencies 78, 16, 30, 35, 125, 31, 20, 50, 80, 3.
Let T = (V, E) be a complete m-ary tree of height h. This tree is called a full m-ary tree if all of its leaves are at level h. If T is a full m-ary tree with height 7 and 279,936 leaves, how many internal vertices are there in T?
Let T be a full m-ary tree with height h and v vertices. Determine h in terms of m and v.
Using the weights 2, 3, 5, 10, 10, show that the height of a Huffman tree for a given set of weights is not unique. How would you modify the algorithm so as to always produce a Huffman tree of minimal height for the given weights?
Let Ll, for 1 ≤ i ≤ 4, be four lists of numbers, each sorted in ascending order. The numbers of entries in these lists are 75, 40, 110, and 50, respectively. (a) How many comparisons are needed to merge these four lists by merging L1 and L2, merging L3 and L4, and then merging the two resulting
Find the articulation points and biconnected components for the graph shown in Fig. 12.44.
Let G = (V, E) be a loop-free connected undirected graph, where V = {a, b, c, . . . , h, i, j}. Ordering the vertices alphabetically, the depth-first spanning tree T for G - with a as the root-is given in Fig. 12.45(i). In part (ii) of the figure the ordered pair next to each vertex v provides
In step (2) of the algorithm for articulation points, is it really necessary to compute low(x1) and low(x2)?
Let G = (V, E) be a loop-free connected undirected graph with v ˆˆ V.(b) If v is an articulation point of G, prove that v cannot be an articulation point of .
If G = (V, E) is a loop-free undirected graph, we call G color-critical if X(G - v) < X(G) for all v ∈ V. (We examined such graphs earlier, in Exercise 19 of Section 11.6.) Prove that a color-critical graph has no articulation points.
Does the result in Lemma 12.4 remain true if T = (V, E') is a breadth-first spanning tree for G = (V, E)?
Prove Lemma 12.3. Let G = (V, E) be a loop-free connected undirected graph with z ∈ V. The vertex z is an articulation point of G if and only if there exist distinct x, y ∈ V with x ≠ z, y ≠ z, and such that every path in G connecting x and y contains the vertex z.
Let T = (V, E) be a tree with |V| = n ≥ 3. (a) What are the smallest and the largest numbers of articulation points that T can have? Describe the trees for each of these cases. (b) How many biconnected components does T have in each of the cases in part (a)?
(a) Let T = (V, E) be a tree. If v ∈ V, prove that v is an articulation point of T if and only if deg(v) > 1. (b) Let G = (V, E) be a loop-free connected undirected graph with |E| ≥ 1. Prove that G has at least two vertices that are not articulation points.
Let G = (V, E) be a loop-free connected undirected graph with |V| ≥ 3. If G has no articulation points, prove that G has no pendant vertices.
For the loop-free connected undirected graph G in Fig. 12.43(i), order the vertices alphabetically.(a) Determine the depth-first spanning tree T for G with e as the root. (b) Apply the algorithm developed in this section to the tree T in part (a) to find the articulation points and biconnected
Answer the questions posed in the previous exercise but this time order the vertices as h, g, f, e, d, c, b, a and let c be the root of T. (a) Determine the depth-first spanning tree T for G with e as the root. (b) Apply the algorithm developed in this section to the tree T in part (a) to find the
Let G = (V, E) be a loop-free undirected graph with |V| = n. Prove that G is a tree if and only if P(G, λ) = λ(λ - 1)n-1.
(a) Let T = {V, E) be a complete 6-ary tree of height 8. If T is balanced, but not full, determine the minimum and maximum values for |V|. (b) Answer part (a) if T = (V, E) is a complete m-ary tree of height h.
The rooted Fibonacci trees Tn, n ‰¥ 1, are defined recursively as follows:(1) T1 is the rooted tree consisting of only the root;(2) T2 is the same as T1 - it too is a rooted tree that consists of a single vertex; and(3) For n ‰¥ 3, Tn is the rooted binary tree with Tn-2 as its
(a) The graph in part (a) of Fig. 12.48 has exactly one spanning tree - namely, the graph itself. The graph in Fig. 12.48(b) has four nonidentical, though isomorphic, spanning trees. In part (c) of the figure we find three of the nonidentical spanning trees for the graph in part (d). Note that T2
Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 12.49. For n ‰¥ 0, let an count the number of spanning trees of G, whereas bn counts the number of these spanning trees that contain the edge {x1, y1}.(a) Explain why an = an-1 + bn.(b) Find an equation that expresses
Let T = (V, E) be a tree where |V| = v and |E| = e. The tree T is called graceful if it is possible to assign the labels {1, 2, 3, .. . , v} to the vertices of T in such a manner that the induced edge labeling - where each edge {i, j} is assigned the label |i - j|, for i, j ∈ {1, 2, 3, . . . ,
For an undirected graph G = (V, E) a subset of I of V is called independent when no two vertices in I are adjacent. If, in addition, I ˆª {x} is not independent for each x ˆˆ V - I, then we say that I is a maximal independent set (of vertices).The two graphs in Fig. 12.50 are
In part (i) of Fig. 12.51 we find a graceful labeling of the caterpillar shown in part (i) of Fig. 12.50. Find a graceful labeling for the caterpillars in part (ii) of Figs. 12.50 and 12.51.
Develop an algorithm to gracefully label the vertices of a caterpillar with at least two edges.
Consider the caterpillar in part (i) of Fig. 12.50. If we label each edge of the spine with a 1 and each of the other edges with a 0, the caterpillar can be represented by a binary string. Here that binary string is 10001001 where the first 1 is for the first (left-most) edge of the spine, the next
For n ‰¥ 0, we want to count the number of ordered rooted trees on n + 1 vertices. The five trees in Fig. 12.52(a) cover the case for n = 3.(a) Performing a postorder traversal of each tree in Fig. 12.52(a), we traverse each edge twice - once going down and once coming back up. When we
A telephone communication system is set up at a company where 125 executives are employed. The system is initialized by the president, who calls her four vice presidents. Each vice president then calls four other executives, some of whom in turn call four others, and so on. (Each executive who does
For n ‰¥ 1, let tn count the number of spanning trees for the fan on n + 1 vertices. The fan for n = 4 is shown in Fig. 12.53.(a) Show that tn+1 = tn + ˆ‘nl=0 tl, where n ‰¥ 1 and t0 = 1. (b) For n ‰¥ 2, show that tn+1 = 3tn - tn-1. (c) Solve the recurrence
(a) Consider the subgraph of G (in Fig. 12.54) induced by the vertices a, b, c, d. This graph is called a kite. How many nonidentical (though some may be isomorphic) spanning trees are there for this kite?(b) How many nonidentical (though some may be isomorphic) spanning trees of G do not contain
Let T be a complete binary tree with the vertices of T ordered by a preorder traversal. This traversal assigns the label 1 to all internal vertices of T and the label 0 to each leaf. The sequence of 0's and l's that results from the preorder traversal of T is called the tree's characteristic
For k ∈ Z+, let n = 2k, and consider the list L: a1, a2, a3, ..., an. To sort L in ascending order, first compare the entries at and al+(n/2), for each 1 ≤ i ≤ n/2. For the resulting 2k-1 ordered pairs, merge sort the ith and (i + (n/4))-th ordered pairs, for each 1 ≤ i ≤ n/4. Now do a
Let G = (V, E) be a loop-free undirected graph. If deg(v) ≥ 2 for all v ∈ V, prove that G contains a cycle.

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