Prove that every simple, triangle-free, planar graph is 4-colorable. (Hint: show first that such a graph has a vertex of degree at most 3.)

Q: 2.)Is K6 planar? Explain

A:

Q: Under which conditions does: a) A cycle-free graph have an Hamiltonian circuit? b) A complete graph…

A: Euler:- 1) A graph is Euler if and only if the degree of every vertex is even.

Q: Prove that in each component of the context of any graph, the number of vertices of an odd degree is…

A: INTRODUCTION: We need to prove the number of vertices of an odd degree is always even.

Q: Prove that the graph with at least 100 vertices and minimum degree at least 99, there exists a cycle…

A: Answer: Given vertices=100  minimum degree at least=99

Q: QI: Give an example of a plane graph that is both 2-colourable(f) and 2-colourable(v).

A: Step 1  The answer is given in the below step

Q: if graph G contains a cycle C, then every MST for G contains some edge of C. How do you prove this?

A: G = (V, E) is an undirected simple graph in which each edge has a distinct weight, and e is a…

Q: If  a complete bipartite graph Kr,s is regular what can you say about r and s?

A: We need to identify r and s in a complete bipartite graph Kr,s.

Q: (a) Draw a connected 3-regular graph with 6 vertices and no loops (b) Show that every 3-regular…

A: Regular graph is a graph where each vertex has same degree. In regular directed graph if must…

Q: Every strongly connected graph is Eulerian. There is a series of lowercase letters (a-z) that…

A: A directed graph is graph, a set of objects that are connected together.

Q: TRUEorFALSE(circle one). There exists a planar 5-regular graph.

A: Answer to the above question is in step2.

Q: Show that every nonincreasing sequence of nonnegative integers with an even sum of its terms is the…

A: To show that every nonincreasing sequence of nonnegative integers with an even sum of its terms is…

Q: (a) What is the smallest expansion cut in a cycle with n nodes. (b) What is the densest subgraph in…

A: See below step for explanation.

Q: either draw a graph with the stated property, or prove that no such graph exists.   A graph on 45…

A: Here in this question we have given a graph with 45 vertices in which 22 vertices have degree 8 and…

Q: Prove that if G is a bipartite graph with no isolated vertices then a(G)=B'(G)

A: if the result follows from three standard results 1) let α(G) be the vertex covering no of G then…

Q: Prove that if a graph with n vertices and medges is planar and bipartite then m≤2n-4.

A: To prove that if a graph with n vertices and m edges is planar and bipartite, then m ≤ 2n - 4, we…

Q: Draw the state space free to find the solution using back tracking principle of chromatic number of…

A: Algorithm:  Create a recursive function that takes current index, number of vertices and output…

Q: Prove that the Petersen graph is not a Hamiltonian graph.

A: Petersen Graph: The Petersen Graph is a cubic graph of 10 vertices and 15 edges. Each vertex on the…

Q: Prove that there exists a graph G with 5 nodes such that both G and G's complement, have chromatic…

A: The least number of colors required to color a graph's vertices (or nodes) so that no two adjacent…

Q: draw all non isophormic,cycle-free,connected graphs having six vertices

A: Given, draw all non isophormic,cycle-free,connected graphs having six vertices

Q: Are the two graphs in the figure isomorphic? If there are isomorphic, label the vertices o show the…

A:

Q: a d b a) Give the vertex set of the G. b) Give the set of edge of G. c) Find the degree of each of…

A: In this question we have been given with the graph and we have to answer for the graph theory…

Q: Is it true or false?  If it is true, include a (short, but clear) argument why it is true, and if it…

A: Given four statements and we need to find out if they are true or false.

Q: • Theorem (Euler): A graph without isolated vertices is Eulerian iff it is connected and every…

A: - The question is to prove that a graph is a unit graph only if it is connected and the degree of…

Q: Draw a simple undirected graph G that has15 vertices, 15 edges.

A: Given:  vertices=15 edges=15

Q: Prove that k3,3 is a non-planer graph.

A: Introduction  We have to show that K3,3  is a non-planer graph.

Q: There exists a 5-regular graph of size 50 such that the wheel graph W4 is an induced subgraph. T/f?

A: The question has been answered in step2

Q: Find a graph with this sequence of degrees (987753210) or prove that it does not exist.

A: degree sequence: 9 8 7 7 5 3 2 1 0   Hand shaking lemma: First arrange the degree sequence in…

Q: Please help with graph theory   Show that every k-regular graph is a balanced graph

A: Regular Graph:- A regular graph is a graph where the degree of each vertex is equal.  K-regular…

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

A: We will prove the statement by induction on the number of vertices, v. Base case: If v = 3, then the…

Q: Prove that a connected planer graph G with n vertices and m edges has exactly (m - n + 2) regions in…

A: We can prove the statement "a connected planar graph G with n vertices and m edges has exactly (m -…

Q: Use induction to prove that a graph which consists of a single path is bipartite

A: Answer :

Q: Determine whether there is a graph with the given conditions. In each case, if such a graph exists,…

A: Degree of sequence: The degree of sequence is used to determine, the number of vertex and the number…

Q: An undirected graph G with only one simple path between each pair of vertices has two vertices of…

A: In question, it is given that: Undirected G graph with only one simple path between each pair…

Q: Show that a graph is 4-vertex-colourable if and only if it is the union of two bipartite subgraphs.

A: The answer of this question is as follows:

Q: 8. The number of edges of the complete graph Ks is.

A: Planar Graph:   A graph is said to be planar if it can be drawn in a plane so that no edge cross.…

Q: (a) What is the smallest expansion cut in a cycle with n nodes. (b) What is the densest subgraph in…

A: Below is the answer to above question. I hope this will be helpful for you...

Q: It is well-known that planar graphs are 4-colorable. However finding a vertex cover on planar graphs…

A: In below given the aprox vertex problem.

Q: You are given a simple graph. Prove that there must be at least two vertices with the same degree.

A: The above question is solved in step 2 :-

Q: Consider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle…

A: HI THEREI AM ADDING ASNWER BELOWPLEASE GO THROUGH IT THANK YOU

Q: Prove that the degree of each face of G is three. Show all of your work and clearly explain, using…

A: given,  vertices 12, edges=30. According to the Euler's  formula:-  V-E+F=2 Where F= number of…

Q: - Use Euler's inequality (m≤3n-6) to prove that each planar graph has at least two vertices of…

A: In general, Euler's inequality states that for a planar graph with n vertices and m edges, m <=…