More Graphs, Subgraphs and Trees Planarity and Coloring

NCSCOS Goals and Objectives Competency Goal 1: The learner will use matrices and graphs to model relationships and solve problems. Objective 1.02: Use graph theory to model relationships and solve problems

Four Color Theorem According to the four-color theorem we learned that any map can be drawn on the surface of a sphere can be colored with four colors or fewer. If this is true, then why does it take more than four colors to color some graphs?

Try this! Try to redraw the graphs so that their edges intersect at the vertices. (Think of the edges as Rubber Bands.) K 4 graphK 5 graph A B C D A B C D E

Planar Graphs It is relatively easy to redraw the K 4 graph so that the edges do not cross, but no matter how hard you try, at least two edges of the K 5 graph will always intersect. A graph that can be drawn so that no two edges intersect except at a vertex is called a planar graph.

Planar Graphs and the Four Color Theorem When regions of a map are represented by vertices of a graph and edges are drawn between vertices when boundaries exist between regions, the resulting graph is planar. In other words, graphs that come from a map in a plane or on a sphere are always planar. Hence, the four color theorem can be stated: Every planar graph has a chromatic number that is less than or equal to four.

Four Colors? Why do some graphs require more than four colors? Planarity is the key. If the graph is not planar, we don’t know how many colors it will take to color it. For example, the K 5 graph and other nonplanar graphs as the one on the next slide.

A K 3, 3 Graph AD BE CF

You Try! Try to draw the K 3, 3 graph without the edges crossing. This graph has another interesting characteristic, other than being nonplanar. It is an example of a bipartite graph. A bipartite graph is a graph whose vertices can be divided into two distinct sets so that each edge of the graph has one vertex in each set.

Bipartite Graphs A bipartite graph is said to be complete if it contains all possible edges between pairs of vertices in the two distinct sets. Complete bipartite graphs can be denoted by K M, N, where M and N are the number of vertices in the two distinct sets.

Another Bipartite Graph Here is another example of a complete bipartite graph. A B C X Y

The K 3, 2 Graph This graph is bipartite since its vertices can be separated into two distinct sets {A, B, C} and {X, Y}. Every edge has one vertex in each set; and all possible edges from one set to the other are drawn. Determining planarity can be difficult for a large graph, and so a theorem resulted.

Kazimierz Kuratowski In 1930 Kazimierz Kuratowski, a Polish mathematician proved a theorem that determines the planarity of graphs. He identified two important graphs, K 5 and K 3, 3 and noted that any graph that has a K 5 or K 3,3 subgraph is not planar.

Subgraphs A graph G’ is said to be a subgraph of graph G is all of the vertices and edges of G’ are contained in G. Kuratowski also noted that the lack of a K 5 or K 3,3 does not guarantee planarity. Actually, 99% of nonplanar graphs of modest size are nonplanar because of a K 3, 3 subgraph rather than a K 5 subgraph.

Example Is the graph below planar? A B C D E F G

Example Explained Upon close observation of vertices A, B, C, D, E, and F a K 3,3 subgraph can be found. Therefore, the graph is nonplanar.

Practice Problems In Exercises 1 through 3 decide whether the graph is planar or nonplanar. If the graph is planar, redraw it without edge crossings A B C D E F AB C D E F

Practice Problems (cont’d) 3. A B C D E F

Practice Problems (cont’d) 4.The graph below is planar. Draw it without edge crossings. A B C D E F GH I

Practice Problems (cont’d) 5.How do you know that it is not necessary to check the graph in Problem 4 for a K 5 subgraph? 6.The complement of a graph G is customarily denoted by G. G has the same vertices as G but its edges are those not in G. The edges of G and G along with vertices from either set would make a complete graph. Draw the complement of the following graph.

Practice Problems (cont’d) A B C D E

Practice Problems 7.The concept of planarity is extremely important to printing circuit boards for the electronics industry. Why? 8.Construct the following bipartite graphs: a.K 2,3 b. K 2,4 9.For each of the following bipartite graphs, list the two distinct sets into which the vertices can be divided:

Practice Problems (cont’d) A B C D E F G P Q N M R S T V X Y W U Z

10.How many edges are in a K 2,3 graph? A K 4, 3 ? A K M,N ? 11.When does a bipartite graph, K M,N have an Euler circuit? 12.Is the graph on the next slide planar? Does it contain K 5 or K 3,3 as a subgraph?

Practice Problems (cont’d) A B C D E F G H I J

Extensions For many years, mathematicians thought that all nonpolar graphs had K 5 or K 3,3 as subgraphs. Graphs such as the one in Exercise 12 are said to be an extention of a K 5 because it was formed by adding a vertex or vertices to the edges of the K 5 graph. Extensions of K 5 and K 3,3 are nonpolar.