Please answer the following questions- Are the following statements true or false??????1. If a graph has a vertex of degree 3, then the chromatic number must be at least 3.2. If the chromatic number of a graph is 8, then the graph contains a subgraph isomorphic to Kg.3. If the chromatic number of a graph is 7, then the graph is not planar.4. If every vertex of a graph has degree 8 or less, then the chromatic number of the graph is at most 8.5. If a graph contains a subgraph isomorphic to Kg, then the chromatic number of the graph is at least 8 What is the chromatic number of each graph?IM1

Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question. The chromatic number of a graph is the minimum number of colors needed to color the vertices such that no two adjacent vertices are of the same color. 1. The statement is: If a graph has a vertex of degree 3, then the chromatic number must be at least 3. If a vertex has degree 3, then the bipartite graph K1,3 is a subgraph. Only two distinct colors are required for the bipartite graph K1,3, since the two parts are each independent sets and can be colored with a single color. If the graph has only 4 vertices, then K1,3 is the required graph. The statement is false. 2. The statement is: If the chromatic number of a graph is 8, then the graph contains a subgraph isomorphic to K8. The graph K8 is a complete graph, that is, each vertex is adjacent to every other vertex. The chromatic number of K8 is 8. If a graph has chromatic number 8, then K8 is its subgraph. The statement is true.3. The statement is: The chromatic number of a graph is 7, then the graph is not planar. If the chromatic number of a graph is 7, then K7 is a subgraph of the given graph. The complete graph Kn for n≥4 is not planar. If the subgraph is not planar, then the entire graph is not planar. The given statement is true. 4. The statement is: If every vertex of a graph has degree 8 or less, then the chromatic number of the graph is at most 8. Assume to the contrary that the chromatic number of the graph is 9. Then the graph contains K9 as its subgraph. This further implies that there are vertices of degree 9, which is a contradiction. Hence, the assumption is false. The given statement is true. 5. The statement is: If a graph contains a subgraph isomorphic to K8, then the chromatic number is of the graph is at least 8. The chromatic number of a complete graph Kn is n. Hence, the chromatic number of K8 is 8. Since K8 is a subgraph of the given graph, the chromatic number is of the graph is at least 8. The statement is true.