Let A be a tree on n ≥ 2 vertices. Determine χ(A) and provide proof. Now let A be the attached graph, which can also be referred to as the Petersen Graph. Determine the chromatic number of this graph and provide proof.

Q: Find the Chromatic Number of K, the complete graph on 17 vertices

A: See the solution below.

Q: 4- 3 C

A: By the theorem of Eulerian graph, a graph is Eulerian graph if and only if degree of each vertex is…

Q: 10) Add one edge to the graph in order to get a regular graph of degree 3.

A:

Q: In each part, either draw a graph with the given specifications or explain why no such graph exists.…

A: Here  means a complete bipartite graph with vertices partition  so that whose graph look like this :

Q: Define Mycielski’s construction. Use this to obtain a graph with chromatic number 4 from K2.

A:

Q: Suppose a (simple) graph G = (V, E) is regular with odd vertex degree. Prove that the number of…

A:

Q: For each graph below, find an Euler trail in the graph or explain why the graph does not have an…

A:

Q: For

A: We use given hint.

Q: Using the representation drawn, prove that the Petersen graph has genus 1. Please explain carefully…

A: To prove that the Petersen graph has genus 1, we need to show that the Petersen graph can be…

Q: The "Friendship Theorem" states the following: Suppose in a group of people we have the situation…

A: The "Friendship Theorem" in graph theory can be stated as: If G is a graph in which any two distinct…

Q: We say that two distinct edges in a graph are adjacent if there is a vertex incident to both (the…

A: Solution

Q: Let G be a graph with 14 vertices, six of which is degree 4. If all of its other vertices has degree…

A:

Q: Given a graph B whose vertices have degree at most 3, show that you need at least 4 colors to color…

A:

Q: Let G be a graph with n vertices and m edges, and let v be a vertex of G of degree k and e be an…

A: Degree of a vertex is defined as that from a vertex how many distinct edges are connected. Number…

Q: Show that every simple graph on n vertices subgraph of K.

A:

Q: for which values of m and n are km,n regular?

A: Given: A simple graph is called regular if every vertex of this graph has the same degree. A regular…

Q: b a

A:

Q: complement.

A:

Q: Question 4. Let K, be the complete graph on n vertices. How many edges does it have? Justify your…

A: Given that, the complete graph kn has n vertices. Number of the edges of this complete graph=nn-12

Q: Given a graph B whose vertices have degree at most 3, show that you need at least 4 colors to color…

A: We will use induction on number of vertices.

Q: Consider a simple graph G with 7 vertices labeled A through G. Suppose that every vertex is…

A:

Q: Let r be a natural number. An r-regular graph is a graph in which all vertices have degree r. Assume…

A:

Q: What is the vertex chromatic number and edge chromatic number of the graph given below

A: This is a problem of Grapg Theory.

Q: Let G be graph with 8 vertices and 10 edges. If every vertex of G has degree 2 or 3, then How many…

A:

Q: Draw a graph with five vertices in which each vertex is of degree 3.

A: