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: Give two properties of a graph that ensure that it possesses an Euler circuit.

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Q: 4. Prove: "Show that the number of edges in a simple planar graph of order n is at most 3n – 6".…

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Q: Let G be a simple connected planaı 13 vertices and 19 edges. Then, the number of faces in the planar…

A: Given that, G is a simple connected planar graph with 13 vertices and 19 edges.

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: Consider a bipartite graph on the set A = {1,2,3, 4, 5, 6} and B = {7,8,9, 10, 11, 12}, there is an…

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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: Suppose a connected planar graph G has 27 vertices. How many edges must G have if there are 16…

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

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

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

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Q: oes the graph K45 (the complete graph on 45 vertices) have an Eulerian Circuit? D Yes O No

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Q: Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex…

A: Given that  The no of vertices = 9 degree of each vertices =3 To show such graph is not possible.