Graph T


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Graph T

  1. 1. Graph Theory through an example of Map Coloring<br />
  2. 2. Look at the map of an Island with 6 regions<br /> A, B, C, D, E, F<br />Figure 1<br />
  3. 3. We want to give <br />a unique color to each region.<br />This can be put in mathematical terms as follows:<br />V = {A, B, C, D, E, F}, <br />C – a set of colors, f: V C<br />We callfa coloring of the map.<br />For obvious reasons, we decide to give distinct colors to two neighboring regions.<br />Iff satisfies this condition, we call it aproper coloring.<br />
  4. 4. Now, two questions arise:<br />What is the minimum number of colors required for a proper coloring?<br />With a certain number of colors we have, in how many ways can a proper coloring be done? <br />For the given map with only 6 regions it may be easy to answer these questions. <br />We can make it easy for any map as follows.<br />
  5. 5. For the given map, <br />represent each region by a point. <br />If two regions are neighboring, <br />draw a line segment joining them.<br />In the resulting figure,<br />we refer to the points asverticesand <br />the line segments asedges.<br />.<br />.<br />.<br />.<br />F<br />D<br />B<br />A<br />.<br />.<br />C<br />E<br />Figure 2<br />
  6. 6. Figure 2 represents what we call agraph in mathematics. <br />Let us have its formal definition:<br />V – a finite set, E – a set of pairs of elements in V<br />G = (V, E) is called a graph.<br />Elements of V are called vertices and <br />those of E are called edges. <br />If {a, b} ε E, we say the vertices a and b are adjacent.<br />Now, the question of coloring <br />the regions of the map <br />becomes that of allotting a color to <br />each vertex of the graph G. <br />Map coloring becomes graph coloring. <br />
  7. 7. In Figure 2<br />the graph is G = (V, E), where<br />V = {A, B, C, D, E, F}<br />E = {AB, AC, BC, BD, BE, CE, DE, DF, EF}, <br />Where AB stands for the edge {A, B}.<br />
  8. 8. Graph Coloring<br />If G = (V, E) is a graph and C is a set of colors, <br />then a function f: V  C is called <br />a coloring of the graph.<br />It is a proper coloring if <br />adjacent vertices get distinct colors.<br />The least number of colors <br />required for a proper coloring is called <br />the chromatic number of the graph, <br />denoted by (G).<br />If  is the number of colors available, <br />the number of possible proper colorings will be a polynomial, denoted by p(G, ) – the chromatic polynomial.<br />
  9. 9. Two Special Cases:<br />G = (V, E), E =  - null graph<br />G = (V, E), E is the set of all pairs of elements in V – complete graph.<br />If |V| = n, the complete graph is denoted by Kn<br />
  10. 10. Examples<br />Null Graph: G = (V, E), |V| = n, E = p(G, ) = n(G) = 1<br />Complete Graph: G = Kn p(G, ) = (n) = ( - 1)( - 2)…( - n + 1)(G) = n<br />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<br />
  11. 11. Union and Intersection of graphs<br />
  12. 12. Example 1<br />Consider the graph G = (V, E), <br />V = {A, B, C, D}, E = { AB, AC, AD, BC, CD}<br />G1 = (V1, E1), V1 = {A, B, C}, E1 = {AB, AC, BC}<br />G2 = (V2, E2), V2= {A, D, C}, E2 = {AD, AC, DC}<br />G is the union of G1 and G2<br />Their intersection is H = ({A, C}, {AC})<br />.<br />.<br />D<br />C<br />.<br />.<br />A<br />B<br />Figure 4<br />
  13. 13. Theorem<br />If G = (V, E) is a graph and G1, G2 are its subgraphs such that G = G1G2, G1G2 = Kn, then<br />
  14. 14. Example 2<br />In Example 1<br />G1 = K3, G2 = K3 <br />G1G2 = K2<br />Therefore,<br />Here p(G, 1) = p(G, 2) = 0. What does this mean?<br />And p(G, 3) = 6<br />So, the chromatic number (G) = 3<br />There are 6 proper colorings possible with 3 colors.<br />
  15. 15. Example 3<br />Now, let us take up the initial problem of map coloring.<br />In the corresponding graph,<br />let V1 = {A, B, C, E}, <br />E1 = {AB, AC, BC, BE, CE}<br />V2 = {B,D,E,F}, <br />E2 = {BD, BE, DE, DF, EF}<br />G1 = (V1, E1), G2 = (V2, E2)<br />G = G1G2, G1G2 = K2<br />
  16. 16. By Example 2, <br />p(G1, ) = p(G2, ) = (-1)(-2)2<br />Also, p(K2, ) = (-1)<br />Thus, (G) = 3, and there are 6 proper colorings possible with 3 colors. <br />.<br />.<br />.<br />.<br />F<br />D<br />B<br />A<br />.<br />.<br />C<br />E<br />
  17. 17. A Proper Coloring of the Map<br />