Graph Theory through

an example of Map Coloring

Look at the map of an Island with 6 regions A, B, C, D, E, F

Figure 1

We want to give a unique color to each region.This can be put in mathematical terms as follows: V = {A, B, C, D, E, F}, C – a set of colors, f: V CWe call f a coloring of the map.For obvious reasons, we decide to give distinct colors to two neighboring regions. If f satisfies this condition, we call it a proper coloring.

Now, two questions arise:1. What is the minimum

number of colors required for a proper coloring?

2. With a certain number of colors we have, in how many ways can a proper coloring be done?

For the given map with only 6 regions it may be easy to answer these questions.

We can make it easy for any map as follows.

For the given map, represent each region by a point. If two regions are neighboring, draw a line segment joining them.In the resulting figure,we refer to the points as vertices and the line segments as edges.B D FA

C EFigure 2

. . . .. .

Figure 2 represents what we call a graph in mathematics. Let us have its formal definition:V – a finite set, E – a set of pairs of elements in VG = (V, E) is called a graph.Elements of V are called vertices and those of E are called edges. If {a, b} ε E, we say the vertices a and b are adjacent.

Now, the question of coloring the regions of the map becomes that of allotting a color to each vertex of the graph G. Map coloring becomes graph coloring.

In Figure 2the graph is G = (V, E), whereV = {A, B, C, D, E, F}E = {AB, AC, BC, BD, BE, CE, DE, DF, EF}, Where AB stands for the edge {A, B}.

Graph ColoringIf G = (V, E) is a graph and C is a set of colors, then a function f: V C is called a coloring of the graph.It is a proper coloring if adjacent vertices get distinct colors.The least number of colors required for a proper coloring is called the chromatic number of the graph, denoted by (G).If is the number of colors available, the number of possible proper colorings will be a polynomial, denoted by p(G, ) – the chromatic polynomial.

Two Special Cases:1. G = (V, E), E = - null graph2. G = (V, E), E is the set of all pairs of elements in V

– complete graph. If |V| = n, the complete graph is denoted by Kn

Examples1. Null Graph: G = (V, E), |V| = n, E =

p(G, ) = n (G) = 1

2. Complete Graph: G = Kn p(G, ) = (n) = ( - 1)( - 2)…( - n + 1)(G) = n

3. Linear Graph: G = (V, E), V = {v1, v2,…,vn}, E = {{vi, vi+1}, i = 1, 2, …, n-1}p(G, ) = ( -1)n-1 (G) = 2

Union and Intersection of graphs

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Example 1Consider the graph G = (V, E), V = {A, B, C, D}, E = { AB, AC, AD, BC, CD}G1 = (V1, E1), V1 = {A, B, C}, E1 = {AB, AC, BC}G2 = (V2, E2), V2= {A, D, C}, E2 = {AD, AC, DC}G is the union of G1 and G2

Their intersection is H = ({A, C}, {AC})

A B

CD

Figure 4

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TheoremIf G = (V, E) is a graph and G1, G2 are its subgraphs such that G = G1G2, G1G2 = Kn, then

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),(n

GpGpGp

Example 2In Example 1G1 = K3, G2 = K3

G1G2 = K2

Therefore,

Here p(G, 1) = p(G, 2) = 0. What does this mean?And p(G, 3) = 6So, the chromatic number (G) = 3There are 6 proper colorings possible with 3 colors.

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)2)(1(),(

Gp

Example 3Now, let us take up the initial problem of map coloring.In the corresponding graph,let V1 = {A, B, C, E}, E1 = {AB, AC, BC, BE, CE}V2 = {B,D,E,F}, E2 = {BD, BE, DE, DF, EF}G1 = (V1, E1), G2 = (V2, E2)G = G1G2, G1G2 = K2

By Example 2, p(G1, ) = p(G2, ) = (-1)(-2)2

Also, p(K2, ) = (-1)

Thus, (G) = 3, and there are 6 proper colorings possible with 3 colors.

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21 )2)(1(),().,(

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GpGpGpFinally

B D FA

C E

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A Proper Coloring of the Map