Let X denote the set of all finite graphs. Consider the following statement:For all finite graphs G, if G is planar and has no loops, then thechromatic number of G is at most 4.(a) Write down the structural part of a proof that proceeds directly.(b) Write down the structural part of a proof that proceeds via the contrapositive.Please note, some of the terms in the statement above were not introduced in ourcourse. You do not need to know what they mean in order to complete the question.

(a) Structural part of a proof that proceeds directly: 1. Assume G is a finite graph that is planar and has no loops. 2. Show that G can be colored with at most 4 colors. 3. Use the fact that any planar graph can be colored with at most 4 colors. 4. Demonstrate that G satisfies the conditions for being colored with at most 4 colors. 5. Conclude that the chromatic number of G is at most 4. (b) Structural part of a proof that proceeds via the contrapositive: 1. Assume G is a finite graph with a chromatic number greater than 4. 2. Show that G is either not planar or has loops. 3. Use the fact that any graph with a chromatic number greater than 4 must contain a subgraph that is not planar or has loops. 4. Demonstrate that G contains such a subgraph. 5. Conclude that if the chromatic number of G is greater than 4, then G is either not planar or has loops. ####

Define planar graphs. Is 𝐾4, the complete graph with 4 vertices, a planar graph? Justify

4. Which of the following is true?(a) A graph may contain no edge and many vertices(b) A graph may contain no edges and no vertices(c) A graph may contain no edges and no vertices(d) A graph may contain no vertices and many edges

A planar graph can have 7 vertices, 9 edges, and 3 faces.Question 2Select one:TrueFalse

If every node u in G adjacent to every other node v in G, A graph is said to be Complete Options true false