Show that if a circuit in a planar graph encloses exactly two regions, each of which has an even number of boundary edges, then the circuit has even length

Q: If G1 and G2 are isomorphic, must they have the same number of vertices? If G1 and G2 have the same…

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Q: Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut…

A: Proof: The proof uses induction on n. The theorem is trivial for n=1. Suppose x and y are separated…

Q: If G1 and G2 are isomorphic, must they have the same number of edges? If G1 and G2 have the same…

A: If G1 and G2 are isomorphic then they must have the same number of edges.But if G1 and G2 have same…

Q: 4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does…

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Q: Does a Hamiltonian path or circuit exist on the graph below?

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Q: Prove that a cycle with even number of vertexes is bipartite graph.

A: A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U…

Q: Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: How many non-isomorphic simple graphspre there with 11 vertices, 18 edges, minimum degree 3, maximum…

A: Given, Number of vertices V = 11 Number of edges E = 18 Minimum degree = 3 Maximum degree = 6 Number…

Q: Prove that a graph is two-colorable (bipartite) if and only if it contains no odd-length cycle

A: Let's assume that G is bipartite first. Odd cycles cannot exist in G because every subgraph of G is…

Q: 29. (a) (b) Find all the spanning trees of the network shown in Fig. 7-30(b). sit to you Find all…

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Q: Consider the graphs K,, K3 and K,, in figure given below. Find and draw an Euler 3,3 (traversable)…

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Q: The most famous theorem in graph theorem is the following: Theorem 1. The vertices of a planar graph…

A: No, it is not true that all graphs with χG<4 are planar. it is known that Peterson graph is not…

Q: A Hamiltonian path in a network is a closed path thatpasses exactly once through each node in the…

A: Given: Travelling salesman problem (T.S.P.) and Hamiltonian path information.We need to explain why…

Q: At a cupcake stall at the Greenwich market, the owner is selling six different types of cupcakes.…

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Q: "If G has a walk that uses all the edges, then G is connected." I know that this statement is false…

A: If G has a walk that uses all the edges, then G is connected Clearly this statement is false For…

Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

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Q: find the number of the isomorphism classes of graphs with 5 vertices and 6 edges which have (i)…

A: Given the a graphs with 5 vertices and 6 edges. We have to find the number of the isomorphism…

Q: Ouestion 3 (a) Suppose that T is a tree with n vertices, two of which have degrees r and s,…

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Q: e say that a graph is k-regular if each of its vertices is the endpoint of exactly k edges. Consider…

A: A k-regular bipartite graph G=X⊔Y, E, where 1≤k≤n. A graph is said to be k-regular if each of its…

Q: (4) Show that it is not possible to define a graph with 5 vertices of degree 3, and 2 vertices of…

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Q: Illustrate the key ideas in the proof that a connected graph has an Euler circuit if and only if…

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Q: Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits…

A: A bipartite graph is a simple graph whose vertices can be partitioned into two sets V1​ and V2​ such…

Q: Suppose two Hamiltonian circuits are considered different if the collections of edges that they use…

A: Hamilton circuit: A circuit where every vertex must be passed only ones. Complete graph: A graph G,…

Q: How can I prove "A graph is k-colourable if and only if it is k-partite.”?

A: As per the question we have to prove that a graph is k-colourable if and only if it is k-partite. To…

Q: Grid Grove is a neighborhood, with houses organized in m rows of n columns. Houses that are closest…

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Q: Prove that every cycle in the bipartite graph have even number of edges.

A: A bipartite graph is a graph which contains two vertex sets. Let G be a bipartite graph and let  X…

Q: The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n >…

A: We need to match each statement (a) - (d) to and a graph (1)-(5) such that the chosen graph…

Q: Prove that every hamiltonian bipartite graph is an equally bipartite

A: Let G be a Hamiltonian bipartite graph with bipartition (X, Y). We want to show that |X| = |Y|.

Q: An outerplanar graph is a graph that admits a planar embedding with all vertices on the boundary of…

A: Given that an outerplanar graph is a graph that admits a planar embedding with all vertices on the…

Q: Figure 1 (a) and Figure 1 (b) show a connected graph of six different vertices.   find the least…

A: Given that : By the definition of the connected graph , If there is a path exists between every…

Q: 2.5 Prove that if a graph has exactly two vertices of odd degree, then there is a path from one of…

A: In graph theory, a graph consists of vertices and edges.The degree of a vertex is the number of…

Q: 2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph…

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Q: 2. (a) Find the number of the isomorphism classes of connected graphs with 5 vertices and 5 edges.…

A: Isomorphism is the one-to-one mapping of connected graphs. Check the number of vertices of the graph…

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: Prove that cycles are bipartite for an even number of vertexes. But they are not bipartite for the…

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Q: Prove that the cartesian product of two eulerian graphs is an eulerian graph. hence 2n-cube Q2n is…

A: We are to prove that the cartesian product of two eulerian graphs is an eulerian graph, hence…

Q: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

Q: A path that uses each edge of a connected graph once and ends at the starting vertex is called 1…

A: A path that uses each edge of a connected grapg once and ends at the same starting vertex is called…

Q: (iv) Give an example of a closed walk of length four in the graph that is not a circuit. (v) Give…

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Q: 8) Consider the directed graph below 1 point Ps P1 P2 PA The number of 2-step connections for this…

A: The adjacency matrix is given by A = 0010100010010100000101000 Then  A2 = 0010100010010100000101000…

Q: Prove that if a graph on n vertices is connected, and has n-1 edges, it must be acyclic.

A: A graph G is said to be a connected graph, if there is a path between any two vertices in that…