GSM MOBILE PHONE NETWORKS AND GRAPH COLORING By, Manjeera Thogarcheti

OUTLINE Problem statement Graph construction Special properties Relation to graph problem Depicting the graph solution references

PROBLEM STATEMENT Mobile phone network area is divided into hexagonal regions or cells. Communication tower connects the mobile phones. Four frequency ranges 850, 900, 1800, 1900 bands. Assign these frequencies in such a way that no two towers should have same frequency ranges.

GRAPH CONSTRUCTION (PLANAR GRAPH) A graph can be drawn on a 2D or 3D plane without crossing the edges i.e., they intersect only at end points.

GRAPH CONSTRUCTION (DUAL GRAPH) Convert the map into dual graph. Put vertex inside each region. If the boundaries are shared, then place an edge between the vertices.

SPECIAL PROPERTIES A dual of planar graph is a plane multi graph. If G is a connected plane graph and if G′ is the dual of G then G is isomorphic to the dual of G′. A finite graph is planar if and only if it does not have K5 or K3,3 as a minor

RELATION TO GRAPH PROBLEM Restriction of four frequency ranges to place the cell towers. Usage of graph coloring concept. Chromatic number = 4 Greedy coloring algorithm.

DEPICTING THE GRAPH SOLUTION Basic Greedy coloring algorithm Let V be the number of vertices in the graph. Color first vertex with first color. Do the following for remaining V-1 vertices Consider the currently picked vertex and color it with the lowest numbered color that has not been used on any previously colored vertices adjacent to it. If all previously used colors appear on vertices adjacent to v, assign a new color to it.

REFERENCES https://www.academia.edu/2479839/Ap plications_of_Graph_Coloring_in_Moder n_Computer_Science http://www.ams.org/journals/tran/1932- 034-02/S0002-9947-1932-1501641-2/S0002- 9947-1932-1501641-2.pdf http://en.wikipedia.org/wiki/Planar_grap h http://www.geeksforgeeks.org/graph- coloring-set-2-greedy-algorithm/

THANK YOU!!

ANY QUERIES??