Let G be a connected graph with 3 or more vertices and such that x(G) = 3. Consider a proper 3-coloring of G with colors, red, blue, and green. Prove that there exists a green node that has both a red neighbor and a blue neighbor.

Q: Prove that in any graph G with n vertices, n ≤ α(G)χ(G)

A: Consider a graph G on n vertices. Let the chromatic number χG=k. Then, k is the minimum number of…

Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: Let G be the (3,4)-complete bipartite graph. (a) List all non-isomorphic subgraphs with 5 vertices…

A: Part(a) Solution:Let be the (3,4) complete bipartite graph K3,4      These are the 11…

Q: In graphing, prove that at least one of G and it's complement, G', is connected.

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Q: Prove that every maximal matching in a graph G has at least a'(G)/2 edges.

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Q: Prove that for every coloring (n colors) of a complete graph on 2n+1 vertices there's a…

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Q: When two graphs G and H are isomorphic, this means: there exists a bijection between the vertices of…

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Q: Prove that every connected graph with m vertices has at least (m-1) edges, for every m>1

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Q: When two graphs G and H are isomorphic, this means: there exists a bijection between the vertices of…

A: Two graphs G and H are said to be isomorphic, if there exist a bijection f between their vertices…

Q: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does  not…

A: To show that if P is a u,v-path in a 2-connected graph G, then G does not necessarily contain a…

Q: Let G be a connected graph that has exactly 4 vertices of odd degree: V1, V2, V3 and V4. Show that…

A: Let G be a connected graph which has 4 vertices of odd degree. The objective is to show that there…

Q: If G is a colored graph in which each vertex is assigned a color. The chromatic number of a graph G,…

A: The chromatic number of a graph χG is the minimum number of colors needed in a graph to color it so…

Q: Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a…

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Q: Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only…

A: Given, Connecting two nodes u and v in a graph G by a new edge creates a new cycle such that if and…

Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Euler cycle.

A: AnswerSuppose we have some simple graph G, with odd n>, 1 nodes, and that this graphhas an Euler…

Q: Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a…

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Q: Draw all non-isomorphic graphs of order n with 1<n<4.

A: As per the question we have to draw all the non-isomorphic graphs of order 1, 2, 3, 4 But as there…

Q: 4. Prove: "The complement of the graph G is an Eulerian graph whenever G is also an Eulerian graph…

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Q: b) Is the following graph isomorphic to Q4? If yes, give an isomorphism from Q4 to this graph.…

A: 2b ) Given graph is Here number of vertices in the given graph is 12 and number of vertices in Q4…

Q: Show that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

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Q: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not…

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Q: 3. Determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism (by showing a…

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Q: Prove that every graph G on {1,... n} which is acyclic and has n 2 edges has exactly 2 connected…

A: Given: To prove the given statement as follows:

Q: Prove: For any graph r with six vertices ror r complementary contains C3

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Q: Let G be a simple graph. Show that the relation R on theset of vertices of G such that uRv if and…

A: Given: A simple graph is G. To prove: That the set of vertices of G is such that uRv if and only if…

Q: Let G be a graph with n vertices. Prove that if G is connected and has n 1 edges, then G is a tree.…

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Q: Pick a set of five vertices of G and draw the subgraph induced by that set of vertices G:

A: Let GV,E be a graph with vertex set V and edge set E.  A vertex-induced subgraph HV1,E1 is a…

Q: Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Diracs…

A: Dirac's theorem Let G = (V,E) be a graph with n vertices in which each vertex has degree at least…

Q: graphing, prove that at least one of G and it's complement, G', is connected.  Draw G and its…

A: If G is connected then nothing have to prove.

Q: (b) Show that if a 2-colorable graph G has exactly two 2-colorings, then any two cycles contained in…

A: Given: Demonstrate that if a 2-colorable graph G has exactly two 2-colorings, then any two cycles in…

Q: Show that every simple graph on n vertices subgraph of K.

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Q: Prove that if G = (V, E) is a connected graph such that |E |V| – 1, then G is a tree. %3D

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Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

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Q: Let G be a connected graph with at least two nodes. Prove that it has a node such that if this node…

A: Consider a connected graph with at least two vertices. choose vertices u,v so that du,v=maximum if…

Q: (!) Let G be a graph whose odd cycles are pair wo odd cycles in G have a common vertex. Prove

A: Given : Graph G has odd cycles that are pair-wise intersecting, i.e, every two odd cycles in G has a…

Q: Let G be an undirected graph with a loop at every vertex.Show that the relation R on the set of…

A: Given: A graph is G is an undirected graph and has a loop at every vertex. To prove: The given…

Q: Draw a single graph G which satisfies all of the following properties. - All vertices of G are of…

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Q: For each of the two directed graphs below,

A: Given, Diagram of two directed graphs and . We have to determined that the given two directed…

Q: Show that if an edge e is in a closed, trail of G, then e is in a cycle of G

A: Use the following definitions, to obtain the solution. A trail is a sequence of vertices and edges…

Q: Give an example of two undirected graphs G1 with nodes {f, s, y, r} and G2 with nodes {t, q, n, p,…

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Q: Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).

A: Since you have asked multiple questions, we will solve the first one for you. If you want any…

Q: 36. и

A: In figure 1.  Degree of the vertex are :  deg(u1)=3, deg(u2)=2 ,deg(u3)=3 , deg(u4)=3 , deg(u5)=3…