(23) There exist some disconnected self-complementary graphs." (24) "For any even integer n ≥ 4, there exists a connected graph of order n such that all the vertices have odd degrees and that there is no bridge." (25) "For any even integer k ≥ 6, if C is an induced subgraph of a graph G and if the diameter of G is less than k/2, then G contains at least three cycles."

No written by hand solution

Graph Complement and Planarity,(a) Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph.  for this problem, you must writecomplete sentences, include all details, show all of your work, and clarify all of your reasoning.

5. Another Proof Consider an undirected, unweighted, simple graph G. Suppose the longest simple path in G has length 10 and goes from node u to node v. Prove that the degree of node u is at most 10.