(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. ####

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.

The Four Color Theorem.

If a graph G is k-colorable and k<n, for any integer n then it is __________

What does the chromatic number of a graph represent?Select one:a. The minimum number of colors needed to color the graph so that no two adjacent vertices have the same color.b. The number of vertices in the graph.c. The number of colors used to color each edge of the graph.d. The total number of possible colorings for a given graph.

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