Suppose that G is an r-regular graph of order n such that both G and its complement G are connected. Is it possible that neither G nor G is Eulerian?

Q: Let G be a connected graph. Prove that G is Eulerian if and only if each of the blocks of G is…

A: The objective of the question is to prove that a connected graph G is Eulerian if and only if each…

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: Prove or disprove: There exist two connected graphs G and H both of order at least 3 and neither of…

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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: Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane…

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Q: Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if…

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Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

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

A: Let v0 and v1 be distinct vertices of a graph G=V,E. Assume that there exists path in G from v0 to…

Q: (6) Let G be a graph such that for any u, v € V(G), there exists a unique (u, v)-path. Prove that G…

A: G is a graph such that for any u, v∈VG, there exists a unique u, v-path. To Prove: G is a tree.

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: Given a graph G = (V,E) its complement G is defined to be (V,E) where E = V{2} \ E. That is: a graph…

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Q: Let G be a connected graph with 3 or more vertices and such that x(G) = 3. Consider a proper…

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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: Let G be a simple graph (no loops or multi-edges) and G' denote the complement of G. It is known…

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Q: Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and…

A: Proof:Sufficiency:Suppose that G is 2-connected and for any vertex v and edge e of G, v and e lie on…

Q: Let G be a graph with n ≥ 3 vertices that has a clique of size n − 2 but no cliques of size n − 1.…

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Q: Let G be a bipartite Prove that a(G) = graph of order n. if and only if G has a perfect matching.

A: Let G be a bipartite graph of order n. We need to prove that  if and only if G has a perfect…

Q: Give a family of graphs of order p such that t(G) = 2.

A: The family of graphs of order p such that γtG=2 are given below.

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

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Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For 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…

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Q: Prove that the graphs G₁ and G₂ are isomorphic. f a e G₁ b d C 1 5 3 4 G₂ 2 6

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Q: The symmetric difference graph of two graphs G1 = (V, E1) and G2 = (V, E2) on the samevertex set is…

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Q: Prove that for any two edges of a 2-connected graph, a cycle exists containing both of them.

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Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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Q: Let G be a connected graph of order n and size n. Prove that G contains a single cycle.

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Q: Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for…

A: If we are confident with our definition we can easily attempt like these questions, G is connected…

Q: Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to…

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Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…