 14.1: Prove or disprove: (a) If G is an Eulerian graph and H is a subdivi...
 14.2: True or False. (a) Every 3regular graph has chromatic number 2. (b...
 14.3: Let G be a plane graph. For each vertex v of G with deg v 3, let e1...
 14.4: The degrees of the ten vertices of a connected graph G are 1, 1, 1,...
 14.5: The degrees of the twelve vertices of a connected graph G are 1, 1,...
 14.6: The minimum degree of a planar graph G with V (G) = {v1, v2, . . . ...
 14.7: Show that the minimum degree of a maximal planar graph of order 4 o...
 14.8: Determine the minimum number of colors needed to color the regions ...
 14.9: If a graph G contains K4 as a subgraph, then _(G) 4. Is the convers...
 14.10: Ten small towns T1, T2, . . . , T10 have decided to apply for a cha...
 14.11: Ten juniors from high school were given the opportunity to visit a ...
 14.12: Among all 5regular graphs, let G1 be one with the smallest chromat...
 14.13: Let G be a nonempty graph and let H be a graph obtained from G by s...
 14.14: Determine the largest positive integer k such that _(H) = _(G) = k,...
 14.15: Determine the chromatic number of the graph G of Figure 14.41.
 14.16: Determine the chromatic number of the graph G of Figure 14.42.
 14.17: A connected plane graph G of order 12 has 10 regions. Suppose that ...
 14.18: Prove or disprove: If every edge of a graph G lies on an odd cycle,...
 14.19: A nonplanar graph G of order 7 has the property that Gv is planar f...
 14.20: Prove or disprove: (a) If G is a graph of order 5 that is not K5, t...
 14.21: The degrees of the vertices of a planar graph are 3, 3, 4, 4, 5, 5....
 14.22: Consider the tree T in Figure 14.43. (a) Draw the complement T. (b)...
 14.23: Determine whether the graph G in Figure 14.44 is planar or nonplanar.
 14.24: Determine whether the graphs G1 and G2 in Figure 14.45 are planar o...
 14.25: Let G be a graph with _(G) = k. Suppose that there is a kcoloring ...
 14.26: Determine the chromatic number of the graph G in Figure 14.46.
 14.27: Let G be a graph with _(G) = k and let a kcoloring of G be given. ...
Solutions for Chapter 14: PLANAR GRAPHS AND GRAPH COLORINGS
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 14: PLANAR GRAPHS AND GRAPH COLORINGS
Get Full SolutionsDiscrete Mathematics was written by and is associated to the ISBN: 9781577667308. Since 27 problems in chapter 14: PLANAR GRAPHS AND GRAPH COLORINGS have been answered, more than 12063 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Chapter 14: PLANAR GRAPHS AND GRAPH COLORINGS includes 27 full stepbystep solutions.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).