4 Let G be a digraph with 2 or more vertices which is strongly connected and in which every node has indegree 1 and outdegree 1. Prove that G is a directed cycle.

Q: Let G be a DAG with exactly one source r and such that for any vertex v there exists a unique…

A: Given G is a directed acyclic graph (DAG).It is also given that G has exactly one source r and for…

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

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Q: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

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Q: Prove on the number of vertices that every graph G with n edges has a bipartite subgraph with at…

A: We will first assume that  P(n) : every graph G with n edges has a bipartite subgraph with atleast…

Q: Find a graph G with seven vertices such that G and its complement both have an Euler cycle.

A: Step:- 1 Euler Cycle:- is a cycle in which an Eulerian trail starts and ends on the same vertex.…

Q: 2. Apply Euler's Theorems and Fleury's Algorithm to determine Euler path and Euler circuits in each…

A: Euler Theorem states that If G is connected graph then G has Euler Circuit if and only if degree for…

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: For every graph G on n vertices, either G or its complement with respect to Kn must be Eulerian…

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Q: Euler cycle.

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

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: If an undirected graph has more than two vertices of odd degree, it does not have an Euler path.

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Q: 1. Prove that a graph is bipartite if and only if it doesn't have any simple circuits of odd order.…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Prove that a bipartite graph has a unique bipartition iff it is connected.

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Q: Determine whether G and H are isomorphic? Also find Euler circuit, Euler trail and Hamiltonian…

A: The isomorphism of graph is given below. To show graph contains  Euler circuit and path we use…

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

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Q: 13. A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd…

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Q: 4. Draw a directed graph using the following binary relation A, using only unidirectional arcs.…

A: The given directed graph is: A=1,3,1,6,2,3,2,8,3,5,3,6,4,1,4,5,5,4,6,3,7,6,7,2,8,3. Therefore the…

Q: Let L be the set of undirected graphs. Prove that L is in NL.

A: Let L be the set of undirected graphs. A undirected graph is bipartite if its nodes can be divided…

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: If every vertex of a directed graph has a loop, then the relation associated with it is : Symmetric…

A: A relation in which every element is related to itself is called reflexive relation. since every…

Q: Let G be a DAG with exactly one source r and such that for any vertex v there exists a unique…

A: Given: G is a DAG(direct Acylic Graph) with exactly once source r. There exist a unique path from r…

Q: 4. Prove: a) In a graph, if two vertices are connected by a walk then they are connected by a path.…

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Q: Prove the following theorem: If the graph G is a tree then there is a unique path connecting any two…

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

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Q: Let G be a pseudograph with 6 vertices. If G is having 3 loops and no parallel or multiple edges,…

A: For the given statement

Q: 2. Prove that the size of every k-chromatic graph is at least (

A: Consider the provided question, According to you we need to solve only (2). (2) We need to prove…

Q: 2. Apply Euler's Theorems and Fleury's Algorithm to determine Euler path and Euler circuits in each…

A: Solution :-

Q: 11. > Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: Let G be a pseudograph with 6 vertices. If G is having 3 loops and no parallel or multiple edges,…

A: No. Of edges in complete graph with n vertices is n(n-1)/2

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

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Q: 1. Prove that in a finite directed graph if every edge has at least one outgoing edge, then the…

A: 1. It is given that the graph is directed and finite.Therefore it has finite vertex and it is given…

Q: Let G be a pseudograph with 6 vertices. If G is having 3 loops and no parallel or multiple edges,…

A: A pseudograph is a non-simple graph in which both graph loops and multiple edges are permitted.…

Q: Prove if n greater than or equal 2 and G is connected with 2n odd vertices then G has n open walks…

A: Hamiltonian Path: If we remove any one edge from a Hamiltonian circuit. then we left with a path.…

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: Draw a subgraph of G that is a tree but not a path. Label the vertices. (c) Find a proper vertex…

A: Subgraph of G that is a tree but not a path is

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: 5.1.7. (-) Construct a graph G that is neither a clique nor an odd cycle but has a vertex ordering…

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Q: Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

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Q: Prove: “The complement of the graph G is an Eulerian graph whenever G is also  an Eulerian graph of…

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Q: Let G be a 2-connected graph. Prove that if G has no even cycles, then G is isomorphic to an odd…

A: k-Connected graphs: A graph is said to be k-connected if it has more than k vertices and remains…

Q: Let G be an odd cycle. Prove that any proper 3-coloring of G using the colors red, green, and blue…

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Q: 5. Are the following directed graphs isomorphic? Provide an isomorphism if they are or an argument…

A: To find an directed isomorphic we should first check if the given graphs are isomorphic or not, if…

Q: Let G be a pseudograph with 6 vertices. If G is having 3 loops and no parallel or multiple edges,…

A: 18 Option (a) is correct.

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: d) Draw the subgraph G – E([{a, b, c, x}]) of G. e) Draw a spanning subgraph of G that is connected…

A: Please find the full answer in nextStep

Q: 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of…

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Q: 7. Prove that if G is a 2-connected, {claw, Pa}-free graph of order at least 13, then G is either a…

A: Given that G is a 2-connected graph. If G is a cycle then there is nothing to prove. Suppose G is…

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…