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Linear Algebra
(a) How many spanning subgraphs are there for the graph G in Fig. 11.27(a)?(b) How many connected spanning subgraphs are there in part (a)? (c) How many of the spanning subgraphs in part (a) have
Find all (loop-free) nonisomorphic undirected graphs with four vertices. How many of these graphs are connected?
Each of the labeled multigraphs in Fig. 11.28 arises in the analysis of a set of four blocks for the game of Instant Insanity. In each case determine a solution to the puzzle, if possible.
(a) How many paths of length 4 are there in the complete graph K7? (Remember that a path such as v1 → v2 → v3 → v4 → v5 is considered to be the same as the path v5 → v4 → v3 → v2 →
For each pair of graphs in Fig. 11.29, determine whether or not the graphs are isomorphic.
Determine |V| for the following graphs or multigraphs G. (a) G has nine edges and all vertices have degree 3. (b) G is regular with 15 edges. (c) G has 10 edges with two vertices of degree 4 and all
For n ∈ Z+, how many distinct (though isomorphic) paths of length 2 are there in the n-dimensional hypercube Qn?
Let n ∈ Z+, with n ≥ 9. Prove that if the edges of Kn can be partitioned into subgraphs isomorphic to cycles of length 4 (where any two such cycles share no common edge), then n = 8k + 1 for some
(a) For n ≥ 2, let V denote the vertices in Qn. For 1 ≤ k ≤ ℓ ≤ n, define the relation R on V as follows: If w, x ∈ V, then w R x if w and x have the same bit (0, or 1) in position k and
If G is an undirected graph with n vertices and e edges, let δ = minv∈V{deg(v)} and let ∆ = maxv∈V{deg(v)}. Prove that δ ≤ 2(e/n) < ∆.
Let G - (V, E), H = (V', E') be undirected graphs with f:V → V' establishing an isomorphism between the graphs, (a) Prove that f-1 ; Vʹ → V is also an isomorphism for G and H. (b) If a ∈ V,
For all k ∈ Z+ where k ≥ 2, prove that there exists a loop- free connected undirected graph G = (V, E), where |V| = 2k and deg(v) = 3 for all v ∈ V.
Prove that for each n ∈ Z+ there exists a loop-free connected undirected graph G = (V, E), where |V| = 2n and which has two vertices of degree i for every 1 ≤ i ≤ n.
Complete the proofs of Corollaries 11.1 and 11.2.
Let He a fixed positive integer and let G = (V, E) be a loop-free undirected graph, where deg(u) > k for all v ∈ V. Prove that G contains a path of length k.
(a) Explain why it is not possible to draw a loop-free connected undirected graph with eight vertices, where the degrees of the vertices are 1, 1, 1, 2, 3, 4, 5, and 7. (b) Give an example of a
(a) Find an Euler circuit for the graph in Fig. 11.44.(b) If the edge {d, e} is removed from this graph, find an Euler trail for the resulting subgraph.
When visiting a chamber of horrors, Paul and David try to figure out whether they can travel through the seven rooms and surrounding corridor of the attraction without passing through any door more
(a) Find the maximum length of a trail in(i) K6(ii) K8(iii) K10 (iv) K2n, n ∈Z+(b) Find the maximum length of a circuit in(i) K6(ii) K8(iii) K10 (iv) K2n, n ∈ Z+
(a) Let G = (V, E) be a directed graph or multigraph with no isolated vertices. Prove that G has a directed Euler circuit if and only if G is connected and od(v) = id(v) for all v ∈ V. (b) A
Let G be a directed graph on n vertices. If the associated undirected graph for G is Kn, prove that ∑v∈V[od(v)]2 = ∑v∈V [id(v)]2.
If G = (V, E) is a directed graph or multigraph with no isolated vertices, prove that G has a directed Euler trail if and only if (i) G is connected; (ii) od(v) = id(v) for all but two vertices x, y
Let V = {000, 001, 010, . . ., 110, 111}. For each four-bit sequence b1b2b3b4 draw an edge from the element b1b2b3 to the element b2b3b4 in V. (a) Draw the graph G = (V, E) as described, (b) Find a
Let G = (V, E) be a connected undirected graph.(a) What is the largest possible value for |V| if |E| = 19 and deg(v) > 4 for all v ∈ V?(b) Draw a graph to demonstrate each possible case in part
Let G = (V, E) be a loop-free connected undirected graph with |V| ≥ 2. Prove that G contains two vertices v, w, where deg (v) = deg(w).
If G = (V, E) is an undirected graph with |V|-n and |E| = k, the following matrices are used to represent G. Let V = {v1, v2, . . . , Define the adjacency matrix A = (aI j)nÃn where aIJ
Determine whether or not the loop-free undirected graphs with the following adjacency matrices are isomorphic.(a)(b) (c)
Determine whether or not the loop-free undirected graphs with the following incidence matrices are isomorphic.(a)(b) (c)
There are 15 people at a party. Is it possible for each of these people to shake hands with (exactly) three others?
Consider the two-by-four grid in Fig. 11.34. Assign the partial Gray code A = {00, 01, 11} to the three horizontal levels: top (00), middle (01), and bottom (11). Now assign the partial Gray code B =
Prove that the three-by-three grid of Fig. 11.34 is isomorphic to a subgraph of the hypercube Q4.
(a) Let G = (V, E) be a loop-free undirected graph, where |V| = 6 and deg(v) = 2 for all v ∈ V, Up to isomorphism how many such graphs G are there? (b) Answer part (a) for | V| -7. (c) Let G1 =
Let G1 = (V1, E1) and G2 = (V2, E2) be the loop-free undirected connected graphs in Fig. 11.42.(a) Determine | V1|, |E1|, |V2| and |E2|.(b) Find the degree of each vertex in V1. Do likewise for each
Let V = {a, b, c, d, e, f}. Draw three nonisomorphic loop-free undirected graphs G1 = (V, E1), G2 = (V, E2), and G3 = (V, E3), where, in all three graphs, we have deg (a) = 3, deg(b) = deg(c) = 2,
(a) Find the number of edges in Q8. (b) Find the maximum distance between pairs of vertices in Q8. Give an example of one such pair that achieves this distance. (c) Find the length of a longest path
(a) What is the dimension of the hypercube with 524,288 edges? (b) How many vertices are there for a hypercube with 4,980,736 edges?
Verify that the conclusion in Example 11.16 is unchanged if Fig. 11.48(b) has edge {a, c} drawn in the exterior of the pentagon.
Can a bipartite graph contain a cycle of odd length? Explain.
Let G = (V, E) be a loop-free connected graph with | V | = v. If | E | > (v/2)2, prove that G cannot be bipartite.
(a) Find all the nonisomorphic complete bipartite graphs G = (V, E), where |V| = 6. (b) How many nonisomorphic complete bipartite graphs G = (V, E) satisfy |V| =n ≥ 2?
(a) Let X = {1, 2, 3, 4, 5}. Construct the loop-free undirected graph G = (V, E) as follows: • (V): Let each two-element subset of X represent a vertex in G. • (E): If v1, v2 ∈ V correspond to
Determine which of the graphs in Fig. 11.69 are planar. If a graph is planar, redraw it with no edges overlapping. If it is nonplanar, find a subgraph homeomorphic to either K5 or K3 3.
Let m, n ∈ Z+ with m ≤ n. Under what condition(s) on m, n will every edge in Km,n be in exactly one of two isomorphic subgraphs of Km,n?
Prove that the Petersen graph is isomorphic to the graph in Fig. 11.70
Determine the number of vertices, the number of edges, and the number of regions for each of the planar graphs in Fig. 11.71. Then show that your answers satisfy Euler's Theorem for connected planar
Let G = (V, E) be an undirected connected loop-free graph. Suppose further that G is planar and determines 53 regions If, for some planar embedding of G, each region has at least five edges in its
Show that when any edge is removed from K5, the resulting subgraph is planar. Is this true for the graph K3,3?
Suppose that G = (V, E) is a loop-free planar graph with | V | = v, | E | = e, and k (G) = the number of components of G. (a) State and prove an extension of Euler's Theorem for such a graph, (b)
Prove that every loop-free connected planar graph has a vertex v with deg (u) ≤ 6.
(a) Let G = (V,E) be a loop-free connected graph with | V | ≥ 11. Prove that either G or its complement must be nonplanar. (b) The result in part (a) is actually true for | V | ≥ 9, but the
(a) Let k Z+, k ¥ 3. If G = (V, E) is a connected planar graph with |V| = v, |E| = e, and each cycle of length at least k, prove that(b) What is the minimal cycle length in
(a) Find a dual graph for each of the two planar graphs and the one planar multi graph in Fig. 11.72.(b) Does the dual for the multigraph in part (c) have any pendant vertices? If not, does this
(a) Find duals for the planar graphs that correspond with the five Platonic solids. (b) Find the dual of the graph Wn, the wheel with n spokes (as defined in Exercise 14 of Section 11.1).
(a) Show that the graphs in Fig. 11.73 are isomorphic.(b) Draw a dual for each graph.(c) Show that the duals obtained in part (b) are not isomorphic.(d) Two graphs G and H are called 2-isomorphic if
Find the dual network for the electrical network shown in Fig. 11.76.
Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and |V| = n, what is |E|?
Let G1, G2 be two loop-free connected undirected graphs. If G1, G2 are homeomorphic, prove that (a) G1, G2 have the same number of vertices of odd degree; (b) G1 has an Euler trail if and only if G2
(a) How many vertices and how many edges are there in the complete bipartite graphs K4,7, K7,11, and Km,n, where m, n, ∈ Z+?(b) If the graph Km 12 has 72 edges, what is m?
Prove that any subgraph of a bipartite graph is bipartite.
For each graph in Fig. 11.68 determine whether or not the graph is bipartite.
Let n ∈ Z+ with n ≥ 4. How many subgraphs of Kn are isomorphic to the complete bipartite graph k1,3?
Let m, n ∈ Z+ with m ≥ n ≥ 2. (a) Determine how many distinct cycles of length 4 there are in Km,n. (b) How many different paths of length 2 are there in Km,n ? (c) How many different paths of
What is the length of a longest path in each of the following graphs? (a) K1,4 (b) K3,7 (c) K7,12 (d) Km,n, where m,n ∈ Z+ with m < n.
How many paths of longest length are there in each of the folio wing graphs? (Remember that a path such as v1 → v2 → v3 → is considered to be the same as the path v3 → v2 → v1.) (a) K1.4
Give an example of a connected graph that has (a) Neither an Euler circuit nor a Hamilton cycle, (b) An Euler circuit but no Hamilton cycle, (c) A Hamilton cycle but no Euler circuit, (d) Both a
(a) Let G = (V, E) be a connected bipartite undirected graph with V partitioned as V1 ∪ V2. Prove that if |V1| + |V2|, then G cannot have a Hamilton cycle. (b) Prove that if the graph G in part (a)
(a) Determine all nonisomorphic tournaments with three vertices. (b) Find all of the nonisomorphic tournaments with four vertices. List the in degree and the out degree for each vertex, in each of
Prove that for n ≥ 2, the hypercube Qn has a Hamilton cycle.
Let T = (V, E) be a tournament with v ∈ V of maximum out degree. If w ∈ V and w ≠ v, prove that either (v, w) ∈ E or there is a vertex y in V where y ≠ v, w, and (v, y), (y, w) ∈ E. (Such
Give an example of a loop-free connected undirected multigraph G = (V, E) such that |V| = n and deg(x) + deg(y) ≥ n - 1 for all x, y ∈ V, but G has no Hamilton path.
Prove Corollaries 11.4 and 11.5. Corollaries 11.4 Let G = (V, E) be a loop-free graph with n (≥ 2) vertices. If deg(u) ≥ (n - l)/2 for all v ∈ V, then G has a Hamilton path. Corollaries 11.5 If
Give an example to show that the converse of Corollary 11.5 need not be true.
Helen and Dominic invite 10 friends to dinner. In this group of 12 people everyone knows at least 6 others. Prove that the 12 can be seated around a circular table in such a way that each person is
Let G = (V, E) be a loop-free undirected ft-regular graph with |V| ≥ 2n + 2. Prove that (the complement of G) has a Hamilton cycle.
For n ≥ 3, let Cn denote the undirected cycle on n vertices. The graph Cn, the complement of Cn, is often called the cocycle on n vertices. Prove that for n ≥ 5 the cocycle Cn has a Hamilton
Let n ∈ Z+ with n ≥ 4, and let the vertex set V' for the complete graph Kn-1 be {v1, v2, v3, . . . , vn-1}. Now construct the loop-free undirected graph Gn = (V, E) from Kn-1 as follows:V = V'
For n ∈ Z+ where n ≥ 4, let V' = {v1 v2, v3, . . . , vn-1} be the vertex set for the complete graph Kn-1Construct the loop-free undirected graph Hn = (V, E) from Kn-1 as follows: V = V' ∪ {u},
Let n = 2k for k ∈ Z+. We use the n k-bit sequences (of 0's and l's) to represent 1, 2, 3, . . . , n, so that for two consecutive integers i, i + 1, the corresponding k-bit sequences differ in
If G = (V, E) is an undirected graph, a subset I of V is called independent if no two vertices in I are adjacent. An independent set I is called maximal if no vertex v can be added to I with I
Let G = (V, E) be an undirected graph with subset I of V an independent set. For each a I and each Hamilton cycle C for G, there will be deg (a) - 2 edges in E that are incident with a
Find a Hamilton cycle, if one exists, for each of the graphs or multigraphs in Fig. 11.84. If the graph has no Hamilton cycle, determine whether it has a Hamilton path.
(a) Show that the Petersen graph [Fig. 11.52(a)] has no Hamilton cycle but that it has a Hamilton path. (b) Show that if any vertex (and the edges incident to it) is removed from the Petersen graph,
Consider the graphs in parts (d) and (e) of Fig. 11.84. Is it possible to remove one vertex from each of these graphs so that each of the resulting subgraphs has a Hamilton cycle?
If n ≥ 3, how many different Hamilton cycles are there in the wheel graph Wn ? (The graph Wn was defined in Exercise 14 of Section 11.1.)
(a) For n ≥ 3, how many different Hamilton cycles are there in the complete graph Kn ? (b) How many edge-disjoint Hamilton cycles are there in K21? (c) Nineteen students in a nursery school play a
(a) For n ∈ Z+, n ≥ 2, show that the number of distinct Hamilton cycles in the graph Kn n is (1/2)(n - 1)! n!. (b) How many different Hamilton paths are there for Kn,n, ft ≥ 1?
Let G = (V, E) be a loop-free undirected graph. Prove that if G contains no cycle of odd length, then G is bipartite.
A pet-shop owner receives a shipment of tropical fish. Among the different species in the shipment are certain pairs where one species feeds on the other. These pairs must consequently be kept in
(a) Determine whether the graphs in Fig. 11.93 are isomorphic.(b) Find P(G, X) for each graph.(c) Comment on the results found in parts (a) and (b).
For n ≥ 3, let Gn = (V, E) be the undirected graph obtained from the complete graph Kn upon deletion of one edge. Determine P(Gn, λ) and x(Gn).
Consider the complete graph Kn for ft ≥ 3. Color r of the vertices in Kn red and the remaining n - r ( = g) vertices green. For any two vertices v, w in Kn color the edge {u, w} (1) red if v, w are
Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 11.94.(a) Determine |V| and |E|.(b) Prove that P(G, λ) = λ(λ - 1)( λ2 - 3
Let G be a loop-free undirected graph, where ∆ = maxu∈V{deg(v)}. (a) Prove that x(G) ∆ + 1. (b) Find two types of graphs G, where x (G) = ∆ + 1.
For n ≥ 3, let Cn denote the cycle of length n. (a) What is P(C3, λ)? (b) If n > 4, show that P(Cn,λ) = P(Pn-1, λ) - P(Cn-1, λ), where Pn-1 denotes the path of length n - 1. (c) Verify that
For n ≥ 3, recall that the wheel graph, Wn, is obtained from a cycle of length n by placing a new vertex within the cycle and adding edges (spokes) from this new vertex to each vertex of the
Let G = (V, E) be a loop-free undirected graph with chromatic polynomial P(G, λ) and |V| = n. Use Theorem 11.13 to prove that P(G, λ) has degree n and leading coefficient 1 (that is, the
Let G = (V, E) be a loop-free undirected graph. (a) For each such graph, where |V| ≤ 3, find P(G, λ) and show that in it the terms contain consecutive powers of λ. Also show that the coefficients
Let G = (V, E) be a loop-free undirected graph. We call G color-critical if x(G) > x(G - v) for all v ∈ V. (a) Explain why cycles with an odd number of vertices are color-critical while cycles with
As the chair for church committees, Mrs. Blasi is faced with scheduling the meeting times for 15 committees. Each committee meets for one hour each week. Two committees having a common member must be
(a) At the J. & J. Chemical Company, Jeannette has received three shipments that contain a total of seven different chemicals. Furthermore, the nature of these chemicals is such that for all 1 ≤ i
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