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  • 1. Graph Theory through an example of Map Coloring
  • 2. Look at the map of an Island with 6 regions
    A, B, C, D, E, F
    Figure 1
  • 3. 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 C
    We callfa coloring of the map.
    For obvious reasons, we decide to give distinct colors to two neighboring regions.
    Iff satisfies this condition, we call it aproper coloring.
  • 4. Now, two questions arise:
    What is the minimum number of colors required for a proper coloring?
    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.
  • 5. 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 asverticesand
    the line segments asedges.
    .
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    F
    D
    B
    A
    .
    .
    C
    E
    Figure 2
  • 6. Figure 2 represents what we call agraph in mathematics.
    Let us have its formal definition:
    V – a finite set, E – a set of pairs of elements in V
    G = (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.
  • 7. In Figure 2
    the graph is G = (V, E), where
    V = {A, B, C, D, E, F}
    E = {AB, AC, BC, BD, BE, CE, DE, DF, EF},
    Where AB stands for the edge {A, B}.
  • 8. Graph Coloring
    If 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.
  • 9. Two Special Cases:
    G = (V, E), E =  - null graph
    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
  • 10. Examples
    Null Graph: G = (V, E), |V| = n, E = p(G, ) = n(G) = 1
    Complete Graph: G = Kn p(G, ) = (n) = ( - 1)( - 2)…( - n + 1)(G) = n
    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
  • 11. Union and Intersection of graphs
  • 12. Example 1
    Consider 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})
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    D
    C
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    A
    B
    Figure 4
  • 13. Theorem
    If G = (V, E) is a graph and G1, G2 are its subgraphs such that G = G1G2, G1G2 = Kn, then
  • 14. Example 2
    In Example 1
    G1 = K3, G2 = K3
    G1G2 = K2
    Therefore,
    Here p(G, 1) = p(G, 2) = 0. What does this mean?
    And p(G, 3) = 6
    So, the chromatic number (G) = 3
    There are 6 proper colorings possible with 3 colors.
  • 15. Example 3
    Now, 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 = G1G2, G1G2 = K2
  • 16. 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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    F
    D
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    A
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    C
    E
  • 17. A Proper Coloring of the Map