Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to color the edges of G red or blue so that every vertex is adjacent to two red edges and two blue edges. Hint: Start by showing that G has a closed Eulerian walk.

Q: Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e…

A:

Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

A:

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: prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are

A: Among the generalized Petersen graphs, having n prism G p,1 on the Durer graph G 6.2 the…

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

A:

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

A:

Q: Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a…

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: Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only…

A: Given, Connecting two nodes u and v in a graph G by a new edge creates a new cycle such that if and…

Q: Use Ore's theorem to prove that any n-vertex graph G for which |E(G)|> (",') +2 is Hamiltonian.

A: answer is in next step

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

A:

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

A:

Q: Prove that every Cayley graph is vertex-transitive.

A:

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

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: Prove that for every odd n ≥ 5 there exist a graph of n+1 vertices n of which have degree 3, and the…

A:

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

A: ANSWER:

Q: Let G be a planar graph in which every face is either a triangle or a pentagon. If G has 12…

A: It is given that Let G be a planar graph in which every face is either a triangle or a pentagon. If…

Q: Let Gbe a simple graph with at least 11l vertices, and let G be its complement. (i) Prove that Gand…

A: Given: G is a simple graph with at least 11 vertices and Ḡ be its complement.

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 a bipartite graph has a unique bipartition iff it is connected.

A:

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

A: Given: To prove the given statement as follows:

Q: Let G be a connected graph that has an Euler tour. Prove or disprove the following statements. If G…

A: Euler Circuit: A closed walk in a graph G that uses every edge exactly once is called an Euler…

Q: 3. A labeled Ks graph with labeled edges appears below. Which edge permutation is induced by the…

A: Solution:

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: Prove the following theorem: If the graph G is a tree then there is a unique path connecting any two…

A:

Q: 2. Prove that the size of every k-chromatic graph is at least (

A: Consider the provided question, According to you we need to solve only (2). (2) We need to prove…

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

A:

Q: Let G be a planar graph. Use Euler's Formula to prove that G must have a vertex of degree 5 or less.…

A:

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Consider the general case, Let G be k-edge connected graph then we shall rove that G is k connected.…

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: Show that every simple graph on n vertices is isomorphic to a subgraph of Kn.

A: We need to prove that every simple graph on n vertices is isomorphic to a subgraph of Kn (complete…

Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

A:

Q: Let G be a connected graph with at least two nodes. Prove that it has a node such that if this node…

A: Consider a connected graph with at least two vertices. choose vertices u,v so that du,v=maximum if…

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

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: Let G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and the other 7…

A: Consider G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and…

Q: Draw a single graph G which satisfies all of the following properties. - All vertices of G are of…

A:

Q: 3. Find the degree of the vertices of the following graphs and verify the handshaking theorem.

A: Degree of a vertex is the number of edges incident on it.

Q: Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. Show…

A: Consider two sets Aand B with N labelled vertices and bipartite graphs between Aand B. Show by…

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

A:

Q: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

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…