What is the minimal number of edges in a simple undirected graph G on 6 vertices which contains an Euler circuit and a vertex of degree at least 3? How many edge disjoint cycles such a minimal graph consists of?

Q: Compute the vertex set V and edge set E of the graph G = (V, E) whose Prufer Code is (3, 3, 3,3,…

A: If the prufer code for a graph has length n, then the graph has n+2 vertices. The prufer code for…

Q: 9. Draw a graph or multigraph without loops with 8 vertices and at least 12 edges that has both an…

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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: 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: 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: 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: Suppose an n-vertex bipartite graph G has exactly k connected components, each of which has two or…

A: Given: The value of n vertex includes the bipartite graph G that has exactly k connected components…

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: 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: Does the graph have an Euler circuit? If the graph does not have an Euler circuit, explain why not.…

A: Given Vertex v0 is connected to vertex v1 and to vertex v9. Vertex v1 is connected to vertex v0, to…

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: Suppose a graph G is connected and haas 106 vertices none of which have degree 1. What is the fewest…

A: We know that The minimum number of edges for undirected connected graph is (n-1) edges. But in this…

Q: Show a Hesse diagram with 5 nodes {1, 2, 3, 4, 5} and 4 edges that has exactly three topological…

A: Given nodes are 1,2,3,4,5 with 4 edges. We know that a topological sorting for a directed acyclic…

Q: e maximum edge count for 4 loop G graph

A: If we have a graph G with n nodes, what is the maximum number of edges in this graph if we allow…

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: How many graphs are there on 5 nodes in which every node has degree at most 2?

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Q: If a planar graph has 18 vertices and 26 edges, how many regions are determined when you draw t…

A: If a connected planar graph G has e edges, v vertices, and r regions, then v-e+r=2. Here,  v=18,…

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: When 3 edges are removed from a graph G without removing any vertices the resulting graph G' is a…

A: here we have given that  3 edges are removed from a graph G without removing any vertices the…

Q: Suppose we add two edges in a tree T to form a graph G. How many cycles can be there in G.

A: When 1 edges are added to a tree,  a cycles are created When 2 edges are added to a tree, either 2…

Q: What is the number of isomorphism classes of graphs with 6 vertices and 3 edges

A: It is known that sum of degree in a graph must be even because sum of degrees is  twice the edges.…

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: 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: Construct a simple graph with vertices F, G, H, I,J that has an Euler circuit and the degree of F is…

A: A graph is said to be simple if it has no loops and parallel edges. Degree of a vertex of a graph is…

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: 2 3

A: For a vertex x, the pair a,b under consideration are such that  i) a≠x,b≠x and  ii) there is a…

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…