Prove or disprove that if G has m edges and n vertices, then the degeneracy of G is at least m

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: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: 11. Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: 4. Show that the edge set of every regular bipartite graph is a union of edge-disjoint perfect…

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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: Prove that every maximal matching in a graph G has at least a'(G)/2 edges.

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Q: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does  not…

A: To show that if P is a u,v-path in a 2-connected graph G, then G does not necessarily contain a…

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: Suppose that a graph G has n vertices, and all vertices have degree greater than or equal to“,.…

A: Let G be a graph with n vertices. Given that all vertices have degree greater than or equal to n-12.…

Q: Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

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Q: Pick a set of five vertices of H and draw the subgraph induced by that set of vertices H: X10

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Q: Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a…

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Q: Let G =(V,E) be a finite graph on n ≥ 3

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Q: Let G = (V, E) be a graph (no loops, no multi-edges, undirected, as always. Let d be the smallest…

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Q: show that if Gòe connected grapht that is mot complete, then contains two vertices u and v such that…

A: This problem is related with graph theory. Here, we have to show that if G is a connected graph that…

Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: (9) Consider a graph with the following set of vertices (v1, v2, v3, v4) and set of edges…

A: In our graph, we have vertices and edges, or both are defined, and to obtain a walk we need to…

Q: G is a graph on n 21 vertices such that both G and its complement have an Euler trail. (Recall that…

A: To do: Draw a simple graph G such that "G is a graph with n≥1 vertices such that both G and its…

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: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not…

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A: To find- Which of the following relations represented by the directed graph is symmetric?   Concept…

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: 13. A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd…

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Q: Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where DI

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Q: The set of all finite connected simple graphs is countable. O true O false

A: Here the given statement is 'the set of all finite connected simple ghaph is countable'

Q: Prove or disprove (by giving a counterexample) that a graph G with n 2 4 vertices and ("5') +3 edges…

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Q: Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in G…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: Suppose that G is a k-regular graph of n vertices, and m edges. The vertex connectivity of G is…

A: Sol

Q: 4. Prove: a) In a graph, if two vertices are connected by a walk then they are connected by a path.…

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Q: Prove or disprove: There exists some n > 1 such that there is a simple graph on n vertices, where…

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Q: 11. > Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: Pick a set of five vertices of G and draw the subgraph induced by that set of vertices G:

A: Let GV,E be a graph with vertex set V and edge set E.  A vertex-induced subgraph HV1,E1 is a…

Q: Suppose G has V vertices and E edges. Let M and m denote, respectively, the maximum and minimum of…

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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: 2. Give an example of a directed graph with infinitely many vertices in which each verter has at…

A: A graph, i.e. a set of vertices or nodes that are connected together, where each edges are directed…

Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

Q: 1. Prove that in a finite directed graph if every edge has at least one outgoing edge, then the…

A: 1. It is given that the graph is directed and finite.Therefore it has finite vertex and it is given…

Q: Prove if n greater than or equal 2 and G is connected with 2n odd vertices then G has n open walks…

A: Hamiltonian Path: If we remove any one edge from a Hamiltonian circuit. then we left with a path.…

Q: (9) Consider a graph with the following set of vertices (v1, v2, v3, v4) and set of edges…

A: definition: A walk is a sequence of vertices and edges of a graph i.e. if we traverse a graph then…

Q: Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible…

A: Please check the next step for details

Q: Prove that in any simple graph G with n vertices and m edges, 2m ≤ n^2 − n.

A: We need to prove that in any simple graph G with n vertices and m edges, 2m≤n2−n

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…

Q: Let G be a n vertices graph ( n 2 3 ), such that d(u) + d(v) 2 n, for every pair of non adjacent…

A: Let us start proving step by step

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

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Q: 1. Let C(G,k) denote the decision problem of whether the undirected graph G = subset of vertices V'…

A: Let C(G,k) denotes the decision problem, Let G is the given graph and k be the subset of the graph…

Q: Prove that if a plane graph G has n vertices, e edges, f faces and k components, then n−e+ f = k +1.

A: Euler’s Theorem: According to Euler's Theorem, V-E+F=2 Where, V is the number of vertices. E is the…