Let G be a connected graph that does not have P4 or C4 as an induced subgraph (i.e.,G is P4, C4 free). Prove that G has a vertex adjacent to all others

Let G be a connected graph that has exactly 4 vertices of odd degree: v1, V2, V3 and V4. Show that there are paths with no repeated edges from v1 to V2, and from v3 to v4, such that every edge in G is in exactly one of these paths.

Let G be a simple graph with 15 vertices and 4 connected components. Prove thatG has at least one component with at least 4 vertices. What is the largest numberof vertices that a component of G have? Compute the maximum number of edgesof G

Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.