 10.5.1E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.2E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.3E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.4E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.5E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.6E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.7E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.8E: In Exercises 1–8 determine whether the given graph has an Euler cir...
 10.5.9E: Suppose that in addition to the seven bridges of Königsberg (shown ...
 10.5.10E: Can someone cross all the bridges shown in this map exactly once an...
 10.5.11E: When can the centerlines of the streets in a city be painted withou...
 10.5.12E: Devise a procedure, similar to Algorithm 1, for constructing Euler ...
 10.5.13E: In Exercises 13–15 determine whether the picture shown can be drawn...
 10.5.16E: Show that a directed multigraph having no isolated vertices has an ...
 10.5.15E: In Exercises 13–15 determine whether the picture shown can be drawn...
 10.5.14E: In Exercises 13–15 determine whether the picture shown can be drawn...
 10.5.18E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.17E: Show that a directed multigraph having no isolated vertices has an ...
 10.5.19E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.20E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.21E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.22E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.23E: In Exercises 18–23 determine whether the directed graph shown has a...
 10.5.25E: Devise an algorithm for constructing Euler paths in directed graphs.
 10.5.24E: Devise an algorithm for constructing Euler circuits in directed gra...
 10.5.27E: For which values of n do the graphs in Exercise 26 have an Euler pa...
 10.5.29E: Find the least number of times it is necessary to lift a pencil fro...
 10.5.26E: For which values of n do these graphs have an Euler circuit?
 10.5.28E: For which values of m and n does the complete bipartite graph Km,n ...
 10.5.31E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.30E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.32E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.33E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.34E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.35E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.38E: Does the graph in Exercise 31 have a Hamilton path? If so. find suc...
 10.5.37E: Does the graph in Exercise 30 have a Hamilton path? If so. find suc...
 10.5.36E: In Exercises 30–36 determine whether the given graph has a Hamilton...
 10.5.39E: Does the graph in Exercise 32 have a Hamilton path ? If so. find su...
 10.5.42E: Does the graph in Exercise 35 have a Hamilton path? If so, find suc...
 10.5.41E: Does the graph in Exercise 34 have a Hamilton path? If so, find suc...
 10.5.40E: Does the graph in Exercise 33 have a Hamilton path? If so. find suc...
 10.5.44E: For which values of n do the graphs in Exercise 26 have a Hamilton ...
 10.5.43E: Does the graph in Exercise 36 have a Hamilton path? If so. find suc...
 10.5.45E: For which values of m and n does the complete bipartite graph Km n ...
 10.5.46E: Show that the Petersen graph, shown here, does not have a Hamilton ...
 10.5.48E: Can you find a simple graph with n vertices with n ? 3 that does no...
 10.5.47E: For each of these graphs, determine (i) whether Dirac’s theorem can...
 10.5.49E: Show that there is a Gray code of order n whenever n is a positive ...
 10.5.50E: Fleury’s algorithm, published in 1883, constructs Euler circuits by...
 10.5.51E: Fleury’s algorithm, published in 1883, constructs Euler circuits by...
 10.5.52E: Fleury’s algorithm, published in 1883, constructs Euler circuits by...
 10.5.53E: Fleury’s algorithm, published in 1883, constructs Euler circuits by...
 10.5.54E: A diagnostic message can be sent out over a computer network to per...
 10.5.55E: Show that a bipartite graph with an odd number of vertices does not...
 10.5.56E: Draw the graph that represents the legal moves of a knight on a 3 ×...
 10.5.57E: Draw the graph that represents the legal moves of a knight on a 3 ×...
 10.5.58E: a) Show that finding a knight’s tour on an m × n chessboard is equi...
 10.5.59E: Show that there is a knight’s tour on a 3 × 4 chessboard.
 10.5.61E: Show that there is no knight’s tour on a 4 × 4 chessboard.
 10.5.60E: Show that there is no knight’s tour on a 3 × 3 chessboard.
 10.5.64E: Show that there is a knight’s tour on an 8 × 8 chessboard. [Hint: Y...
 10.5.62E: Show that the graph representing the legal moves of a knight on an ...
 10.5.63E: Show that there is no reentrant knight’s tour on an m × n chessboar...
 10.5.65E: The parts of this exercise outline a proof of Ore’s theorem. Suppos...
 10.5.66E: Show that the worst case computational complexity of Algorithm 1 fo...
Solutions for Chapter 10.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 10.5
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 66 problems in chapter 10.5 have been answered, more than 199000 students have viewed full stepbystep solutions from this chapter. Chapter 10.5 includes 66 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.