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One of the necessary conditions for any graph G to be Hamiltonian,is that w(G-S)≤|S|,for any subgraph of G Let's go about this using proof by contradiction by assuming that a Hamiltonian graph is not 2 connected If this were so then there exists a vertice v such that when v is removed from G,G is no longer connected ,that is G has more than one connected component This vertice is a subgraph with |S|=1 But since there's more than one connected component in G-S,w(G-S)>1 This violates the necessary condition w(G-S)≤|S| Thus G has to be 2-connected for the graph to be Hamiltonian
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If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is...
If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is...
If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is Hamiltonian. (Recall that a graph G is H-free if G does not contain an isomorphic copy of H as an induced subgraph.) FIGURE 1. The graphs K13 and K13 + e
If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is Hamiltonian. (Recall that a graph G is H-free if G does not contain an isomorphic copy of...
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by d...
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on...
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are e...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 1) There exist exactly one path between any of two vertices u, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G
3. Given graph...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
47.21. What does it mean for two graphs to be the same? Let G and H be graphs. We say th G is iso...
please throughly explain each step.47.21. What does it mean for two graphs to be the same? Let G and H be graphs. We say th G is isomorphic to H provided there is a bijection f VG)-V(H) such that for all a, b e V(G) we have a~b (in G) if and only if f(a)~f (b) (in H). The function f is called an isomorphism of G to H We can think of f as renaming the vertices of G...
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e....
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e. Suppose G has n vertices and m edges. Let E’ be a given subset of the edges of E such that the edges of E’ do not form a cycle. (E’ is given as part of input.) Design an O(mlogn) time algorithm for finding a minimum spanning tree of G induced by E’. Prove that your algorithm indeed runs in O(mlogn) time. A minimum...
What does it mean for two graphs to be the same? Let G and H be...
What does it mean for two graphs to be the same? Let G and H be graphs. We Say that G is isomorphic to H provided there is a bijection f : V(G) rightarrow V(H) such that for all a middot b epsilon V(G) we have a~b (in G) if and only if f(a) ~ f(b) (in H). The function f is called an isomorphism of G to H. We can think of f as renaming the vertices of G...
If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is Hamiltonian. (Recall that a graph G is H-free if G does not contain an isomorphic copy of H as an induced subgraph.) FIGURE 1. The graphs K13 and K13 + e
If G is a 2-connected graph that is both K1,3-free and (K1,3+e)-free, prove that G is Hamiltonian. (Recall that a graph G is H-free if G does not contain an isomorphic copy of...
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 1) There exist exactly one path between any of two vertices u, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G
3. Given graph...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
What does it mean for two graphs to be the same? Let G and H be graphs. We Say that G is isomorphic to H provided there is a bijection f : V(G) rightarrow V(H) such that for all a middot b epsilon V(G) we have a~b (in G) if and only if f(a) ~ f(b) (in H). The function f is called an isomorphism of G to H. We can think of f as renaming the vertices of G...
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