A connected multigraph has an euler path if and only if it has Every vertex has even degree Every vertex has odd degree None Exactly two vertices of odd degree

Q: Is hypercube graph Q4 planar? If not can you provide any bounds on it crossing number?

A: No, hypercube graph Q4 is not a planner graph.

Q: How many nonisomoprhic connected planar graphs (V, E) are there with v = 6 and e = 10 assuming that…

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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: 9. Draw two simple connected undirected graphs, each with six nodes, with every node of degree 3,…

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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: (b) a)

A: (a) The new Eularian graph is shown in the figure given in step 2.

Q: A connected graph has 80 even vertices and two odd vertices Determine whether the graph has an Euler…

A: Using Euler's theorem,  A graph has an Euler circuit if and only if degree of each vertex is even.…

Q: If a simple graph Gis having 12 vertices and 9 edges then its complement graph G will have O 12…

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

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Q: Show the results of inserting the keys C, R, A, U,J,S, F,T,0,P,M,L,N,W,Q in order into an empty…

A: Given: V={1,2,3,4,5,6,7,8,9}. and edges E(V)={e1=(1,3),e2=(2,4),e3=3,6 ,e4=7,8,…

Q: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

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Q: Total number of hamiltonian circuits of the complete graph K, is 5 36 circuits 24 circuits 12…

A: Since every permutation of the 5 vertices determines a Hamiltonian cycle, but each cycle is counted…

Q: Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1)…

A: Note: We will be solving Q. 1 only as asked.Given that  and  are two sets of  labeled…

Q: Draw an undirected graph that has exactly 11 edges and at least 5 vertices, in which two of the…

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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: Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd length.  Is…

A: if G is bipartite with vertex sets V1 and V2 every step along a walk takes you either from V1 to V2…

Q: Draw an Eulerian graph which has 5 vertices having one vertex with degree 4 and other vertices with…

A: Given: To draw the Eulerian graph which has 5 vertices having one vertex with a degree 4 and the…

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: A K- regular graph is a graph where every vertex has degree K

A: Consider the given information. A graph is hamiltonian if it visits every vertex exactly once except…

Q: In other words, B(n) = Σk-1 S(n, k). Prove the following B(n) = Σ (1) B(i) i=0

A: Given set partition of B(n) = [1,.....,n]. We have to prove B(n) = ∑i=0nniB(i)

Q: Find the number of spanning trees that can be made of this graph using Kirchhoff's Theorem

A: The total number of spanning trees are given by using Kirchhoff's Theorem

Q: How many Euler paths are there in this graph?

A: The objective is to find the Euler path in the graph

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: 2) What is the maximum number of edges possible for any bipartite graph between A and B?

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Q: Show that any connected simple planar graph on n vertices must have at most n/2 vertices of degree…

A: Consider the equation

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: G1 G2 G3 G4 Which of the given graphs contains an Euler path?

A: Given three graphs G1,G2,G3 & G4.

Q: Which of the following statement/s is/are true or false? * Statement I: With Hamilton cycle, every…

A: Hamilton cycle is a closed path which visits every vertex exactly once. So some edge may not be used…

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: Every simple 3-connected planar graph has a unique planar embedding. Please show that why and gıve…

A: Take a 3-connected planar graph G. We denote the mirror image of G by G-1.  Let G' be some other…