Introductory Combinatorics
5th Edition
ISBN: 9780136020400
Author: Richard A. Brualdi
Publisher: Prentice Hall
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Chapter 12, Problem 27E
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To prove: The planar graph G has at least two vertices whose degree are at most 5.
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Chapter 12 Solutions
Introductory Combinatorics
Ch. 12 - Prob. 1ECh. 12 - Prob. 2ECh. 12 -
Prove that the chromatic polynomial of a...Ch. 12 -
Prove that the chromatic number of a cycle graph...Ch. 12 - Prob. 5ECh. 12 -
Prove that a graph with chromatic number equal to...Ch. 12 - Prove that the greedy algorithm always produces a...Ch. 12 - Let G be a graph of order n ≥ 1 with chromatic...Ch. 12 - Let G be a graph of order n whose chromatic...Ch. 12 - Prob. 10E
Ch. 12 -
Prove that the chromatic polynomial of the graph...Ch. 12 -
What is the chromatic number of the graph...Ch. 12 - Prob. 13ECh. 12 -
Prove that the chromatic polynomial of a cycle...Ch. 12 - Prob. 15ECh. 12 -
Prove that the polynomial k4 − 4k3 + 3k2 is not...Ch. 12 - Prob. 17ECh. 12 -
Give an example of a planar graph with chromatic...Ch. 12 - A plane is divided into regions by a finite number...Ch. 12 - Prob. 22ECh. 12 - Prob. 23ECh. 12 - Prob. 26ECh. 12 - Let G be a planar graph of order n ≥ 2. Prove that...Ch. 12 - Find a solution to the problem of the 8 queens...Ch. 12 - Prob. 33ECh. 12 - Prob. 34ECh. 12 - Prob. 36ECh. 12 - Prob. 37ECh. 12 - Prob. 38ECh. 12 - Prob. 45ECh. 12 - Prove that the edge-connectivity of Kn equals n –...Ch. 12 - Prob. 53ECh. 12 - Prob. 54ECh. 12 - Prob. 55ECh. 12 - Determine the edge-connectivity of the complete...Ch. 12 - Prob. 63E
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- Using Karnaugh maps and Gray coding, reduce the following circuit represented as a table and write the final circuit in simplest form (first in terms of number of gates then in terms of fan-in of those gates). HINT: Pay closeattention to both the 1’s and the 0’s of the function.arrow_forwardRecall the RSA encryption/decryption system. The following questions are based on RSA. Suppose n (=15) is the product of the two prime numbers 3 and 5.1. Find an encryption key e for for the pair (e, n)2. Find a decryption key d for for the pair (d, n)3. Given the plaintext message x = 3, find the ciphertext y = x^(e) (where x^e is the message x encoded with encryption key e)4. Given the ciphertext message y (which you found in previous part), Show that the original message x = 3 can be recovered using (d, n)arrow_forwardTheorem 1: A number n ∈ N is divisible by 3 if and only if when n is writtenin base 10 the sum of its digits is divisible by 3. As an example, 132 is divisible by 3 and 1 + 3 + 2 is divisible by 3.1. Prove Theorem 1 2. Using Theorem 1 construct an NFA over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}which recognizes the language {w ∈ Σ^(∗)| w = 3k, k ∈ N}.arrow_forward
- Recall the RSA encryption/decryption system. The following questions are based on RSA. Suppose n (=15) is the product of the two prime numbers 3 and 5.1. Find an encryption key e for for the pair (e, n)2. Find a decryption key d for for the pair (d, n)3. Given the plaintext message x = 3, find the ciphertext y = x^(e) (where x^e is the message x encoded with encryption key e)4. Given the ciphertext message y (which you found in previous part), Show that the original message x = 3 can be recovered using (d, n)arrow_forwardFind the sum of products expansion of the function F(x, y, z) = ¯x · y + x · z in two ways: (i) using a table; and (ii) using Boolean identities.arrow_forwardGive both a machine-level description (i.e., step-by-step description in words) and a state-diagram for a Turing machine that accepts all words over the alphabet {a, b} where the number of a’s is greater than or equal to the number of b’s.arrow_forward
- Compute (7^ (25)) mod 11 via the algorithm for modular exponentiation.arrow_forwardProve that the sum of the degrees in the interior angles of any convex polygon with n ≥ 3 sides is (n − 2) · 180. For the base case, you must prove that a triangle has angles summing to 180 degrees. You are permitted to use thefact when two parallel lines are cut by a transversal that corresponding angles are equal.arrow_forwardAnswer the following questions about rational and irrational numbers.1. Prove or disprove: If a and b are rational numbers then a^b is rational.2. Prove or disprove: If a and b are irrational numbers then a^b is irrational.arrow_forward
- Prove the following using structural induction: For any rooted binary tree T the number of vertices |T| in T satisfies the inequality |T| ≤ (2^ (height(T)+1)) − 1.arrow_forward(a) Prove that if p is a prime number and p|k^2 for some integer k then p|k.(b) Using Part (a), prove or disprove: √3 ∈ Q.arrow_forwardProvide a context-free grammar for the language {a^ (i) b^ (j) c^ (k) | i, j, k ∈ N, i = j or i = k}. Briefly explain (no formal proof needed) why your context-free grammar is correct and show that it produces the word aaabbccc.arrow_forward
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