Let G be a connected graph that has exactly 4 vertices of odd degree: V1, V2, V3 and V4. Show that there are paths with no repeated edges from v1 to V2, and from v3 to v4, such that every edge in G is in exactly one of these paths.

Q: Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7…

A: Given - G be a simple graph with 11 vertices, each of degree 5 or 6 To Prove - G has at least 7…

Q: 29. Let G be a graph with vertex set VG and edge set Ec. Let V C VG and EC EG. Is there always a…

A:

Q: Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets…

A: Let us prove above result

Q: 14. Let G be the graph with vertices v1, v2, and v3 as shown below. What is the number of walks of…

A: Walk :- A walk in graph is a sequence of vertices and edges.

Q: Show that in every simple graph there is a path from every vertex of odd degree to some other vertex…

A: we have to prove that In every simple graph there is a path from every vertex of odd degree to some…

Q: Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits…

A: A bipartite graph is a simple graph whose vertices can be partitioned into two sets V1​ and V2​ such…

Q: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

A: We have to prove :       If u is a vertex of odd degree in a graph, then there exists a path from…

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

A: Consider the equation

Q: Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a…

A: First, we need to understand the problem. We are given two paths P1 and P2 of maximum length in a…

Q: The Euler's formula v - e + f = 2 holds for all connected planar graphs. What if the graph is not…

A: The formula for multiple connected component c is, v +f = e + c + 1

Q: A graph X is called self-complementary if it isomorphic to its complement. For a self-complementary…

A:

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: prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

A:

Q: Complete a path straight at vertex a that ends at vertex c.

A: A complete path is defined when all the vertices are passed through and none of the smaller paths…

Q: Consider the following graph, G: 9 4 7 2 5 8 6 10 3

A: Given a graph.

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A:

Q: Suppose a graph G has the following Harniltonian Cycle: BAGDFHEJKXB a. How many vertices does G…

A:

Q: Consider a graph G with k vertices and m edges. Show that G has at least one vertex of degree at…

A: To prove that a graph G with k vertices and m edges has at least one vertex of degree at most 2, we…

Q: Let G be a planar graph on 12 vertices with 24 edges in which all faces are bounded by cycles of…

A: G is a planar graph.Number of vertices  V=12 , Number of edges  E=24. Let F be the number of faces.…

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

A: Consider the given set's, A=1,2,3,4,5,6 and B=7,8,9,10,11,12 Here there is an edge between x∈A and…