Suppose G is a simple graph with n >1 vertices such that every vertex has degree at least (n – 1)/2. Prove that G must be connected. (Hint: you may wish to prove this by induction on n).

Q: Prove that in any graph G with n vertices, n ≤ α(G)χ(G)

A: Consider a graph G on n vertices. Let the chromatic number χG=k. Then, k is the minimum number of…

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

A:

Q: Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3.…

A: Step 1: Consider the provided question, (17) (i)

Q: 5. Prove that every graph of order 6 with chromatic number 3 has at most 12 eges.

A: Solution :- As per guidelines.i solve 5th number.

Q: Let G be a graph of order n and let k be an integer with 1≤k≤n−1.Prove that if every vertice of G…

A:

Q: A simple graph G on n vertices with m edges is overfull if m> [n/2]A. Show that if G is overfull,…

A: Given that: A simple graph G on n vertices with m edges is overfull if m>n2△. The objective is to…

Q: Let H be a graph If H has at least one vertex of degree greater than zero then there are two of even…

A: Given a graph representing H. Statement: If H has at least one vertex of degree greater than zero…

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

A:

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

A:

Q: Prove that for every coloring (n colors) of a complete graph on 2n+1 vertices there's a…

A:

Q: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

A:

Q: Prove on the number of vertices that every graph G with n edges has a bipartite subgraph with at…

A: We will first assume that  P(n) : every graph G with n edges has a bipartite subgraph with atleast…

Q: Suppose m and n are integers that are not zero together. Show that there is not always a graph with…

A: For a graph we know that,        Sum of degree of vertices = 2 x Number of edges.     => Sum of…

Q: (d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even…

A:

Q: Let A and B each be sets of N labeled as vertices, and consider the bipartite between A and B 1.…

A: Given: A, B be any sets of N labeled vertices in each. Bipartite between the set A and B. To find:…

Q: 4.1.3. (-) Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then…

A: Given: Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then G…

Q: Let G = (V, E) be a connected graph. Use induction to show that if G contains exactly |G| = |V|.…

A: Suppose G=(V,E) be a connected graph, also let that G contains one cycle. Then we show that G=V

Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Consider the 4-vertex graph G associated with the Seven Bridges of Königsberg problem from the…

A:

Q: Suppose G is a simple graph with n > 1 vertices such that every vertex has degree at least (n –…

A: Let G be the simple graph with the vertices n≥1 and every vertex has the degree of at least n-12.…

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

A:

Q: Draw all non-isomorphic graphs of order n with 1<n<4.

A: As per the question we have to draw all the non-isomorphic graphs of order 1, 2, 3, 4 But as there…

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

A:

Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

A:

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

A: Given  be a graph with m edges and n vertices. Required to prove that the degeneracy of G is at…

Q: Q10. Using Corollary 3.1.24 of the book, to prove that a graph G is bipartite if and only if a(H) =…

A: Given: α(H)=β'(H) To find: Isolated vertices

Q: 2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a)…

A: The objective is to give an example of a graph with at leas four vertices, or prove that none…

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

A:

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: 2. Prove the following and make a sketch. a) Prove that every Eulerian graph of odd order has three…

A: 1. Since G is connected, every vertex has positive degree. Since G has an Eulerian cycle, every…

Q: Prove that the order of a k-cycle in S, is k.

A: Suppose σ is a k-cycle in the group Sn. then we follow that σ is a permutation of length k in which…

Q: If J has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is a problem of Graph Theory.

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

A:

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

A:

Q: Suppose G is a connected graph with n > 1 vertices and n − 1 edges. Prove that G has a vertex of…

A:

Q: 4 Let G be a digraph with 2 or more vertices which is strongly connected and in which every node has…

A: Given that G be a digraph with 2 or more vertices which is strongly connected and in which every…

Q: Let Xn be a the number of graphs with vertex set [2n], Every vertex has a degree of 1. What is the…

A: For each Xn, we count graphs that satisfy two conditions: The graph must have a vertex set of 2n…

Q: 2. Suppose G is (2k + 1)-regular for graph some k > 1. Prove that if G contains a perfect matching,…

A: A graph G is perfect, if χ(H) = ω(H) holds for every induced subgraph H of G.   a k-factor is a…

Q: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

A:

Q: 2. Prove: "The size of every bipartite graph of order n is at most ". Provide an 4 illustration and…

A: Definition: A simple graph whose set of vertices can be partitioned into two sets V1 and V2 such…

Q: A (general) bipartite graph G is a simple graph whose vertex set can be partitioned into two…

A:

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: O For odd values of n>1, Kn is Eulerian. O For odd values of na 1, Kn has an Eulerian trail. ) K2…

A: The above statement is true as “A graph is Eulerian if and only if the degree of each vertex is…

Q: A graph is called d-regular if every vertex has a degree of d. Let k and n be odd numbers, and prove…

A: Here we have to know the basic concept or the basic theorem before answering the question  THEOREM :…

Q: Let Gbe a simple planar graph containing no triangles. (i) Using Euler's formula, show that G…

A: Let G be a simple planar graph containing no triangles.

Q: 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of…

A:

Q: =. Find the smallest 3-regular graph G such that (G) 1.

A:

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

A:

Q: 3.1.7. (-) Use Corollary 3.1.24 to prove that a graph G is bipartite if and only if a(H) = B'(H) for…

A: To prove A graph G is bipartite if and only one α(H)=β'(H) for every subgraph H of G with no…