Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G' obtained from G by deleting the edge e (and keeping all other edges and vertices) is also connected. Show that removing one edge from a connected graph where all vertices have even degree results in a graph that is con- nected.

Q: 1. Is there a connected graph whose degree are: a. 5, 3, 2, 2, 2, 1, 1

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

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Q: If the edges in a path are distinct, it is called:a) simple pathb) simple graphc) length of the…

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Q: Prove that there must be at least two vertices with the same degree in a simple graph

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Q: 2. Show that any graph with at most 8 vertices is planar.

A: Topic:- planer graph

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: Discrete math. Graph theory

A: It is required to draw al, the pairwise non-isomorphic graphs with exactly 5 vertices and 4 edges.

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: (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is…

A: (d)  To prove that the pair of graphs are not isomorphic by showing that there is        a property…

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

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Q: on Find the largest number of paths from 4 to B that do not have any edges in common.

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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: 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: Here are two 3-regular graphs, both with six vertices and nine edges. If they are isomorphic, give…

A: Given ,two graphs are G & HConsider , 3 vertices v1,v2,v3     & 3 edges     e1 , e2  &…

Q: Suppose a (simple) graph G = (V, E) is regular with odd vertex degree. Prove that the number of…

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Q: An edge is called a bridge if the removal of the edge increases the number of connected components…

A: We draw a graph without loop, find number of bridges by removing edges.

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

A: There are multiple questions here.   Question (1) Prove that any graph has at least two vertices…

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: 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: 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: Prove that a graph with at least three vertices is connected if and only if at least two of the…

A: A graph G is a pair of sets V and E where the elements of the non empty set V are called the…

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

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

A: To prove: Any graph has at least two vertices with the same degree. A complete bipartite graph has…

Q: 9. Draw all the non-isomorphic self complementary graphs on four vertices.

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Q: Use the pigeonhole principle to prove that a graph has at least two vertices with the same degree

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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: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: We say that two graphs are distinct if and only if these two graphs are not isomorphic. How many…

A: Given: Two graphs are distinct if and only if these two graph are not isomorphic. To find: How many…

Q: Draw a directed graph with as few edges as possible that is strongly-connected, has 8 vertices of…

A: Strongly connected directed 8 vertices one has in-degree 3 two have cut degree 3 rest have samller…

Q: True or False: The edge set of every closed trail in a graph can be partitioned into edge sets of…

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Q: The number of non-isomorphic 2-regular graphs on 11 vertices is ____.

A: Consider the equation

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

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Q: Does a 3 regular graph on 14 vertices exist what can you say on 17 vertices

A: graph  problem

Q: Prove that the dual to Eulerian planar graph is bipartite.

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Q: Compute the cycle index for the number of non-equivalent graphs with 6 vertices. How many are there?

A: Given:-                 Compute the cycle index for the number of non-equivalent graphs with 6…

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 every bipartite graph with an odd number of vertices does not have a Hamilton cycle/circuit

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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: Does there exist a graph G with 28 edges and 12 vertices each of degree 3 or 4?

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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: Suppose five players are competing in a tennis tournament. Each player needs  to play every other…

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Q: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

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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: Suppose a graph G contains two distinct paths from a vertex u to a vertex v. Show that G has a…

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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: 4. Draw every simple graph with five vertices and four edges (isolated vertices are allowed) up to…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Discrete math. Graph theory

A: Isomorphic graphs are the types of graph which has exactly same number of vertices and connected in…

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…