GRAPH COLORING AND ALLOTTING GATES FOR FLIGHTS PRESENTED BY AMUKTAMALYADA REDDY JAMBULA

OBECTIVES Problem statement Estimation of real-world problem Relating real world problem to graph theory Definitions of graph theory Special properties Problem solution Steps to Solve Summary

P ROBLEM STATEMENT :  Flights needed clear flow in airports, not to face any problems during their landing, take off and waiting time.  If any new flight tries to land unexpectedly then authorities will not be able to land flight on track which there is already other flight scheduled over there.  Due to the delays in landing and other technical problems, not only one flight timings will get effect other flights will also be delayed in arriving their destination.

ESTIMATION OF REAL WORLD PROBLEM We can see in figure(1) that flights in an airport are on track. There are many flights where they may Face problem intake off or land easily. This gives us visual explanation that there may occur traffic jam because of the arrangement of flights. figure(1)

P ROBLEM IN REAL WORLD APPLICATION For example consider EIGHT flights and assign them with alphabets in order to identify them easily A,B,C,D,E,F,G,H. Flight timings without any graph construction gives us following. Flight : Time duration A : 2-4 B : C : 2-8 D : E : 6-18 F : 4-10 G : H : For example here we can see that at one time that is at 12 we can see many flights are in airport and will cause time conflict between flights.

R ELATING REAL WORLD PROBLEM TO GRAPH PROBLEM In this we should use coloring graph such a way that no two flights with time conflict should not have same color. Construct in a way that the number of colors should be less, which may solve other problems like space and cost of airport.

P ROBLEM SOLUTION  This can be solved by allotting GATES to fights by scheduling flights with time intervals without time conflicts with other flight.  Here we can allot different flight timings in different tracks of airports by using graph theory.  Flights which come on same time should assign different gates in terminals, this avoid delays for other flights.

P ROBLEM SOLUTION :  We can give proper solution for gates by taking “set of flights as Vertices and time as Edges”.  Chromatic number usage gives us minimum gates in airport which leads to cost reduction and space of airport.  In this we take timings as primary and based on that we will allot which flight should goes to which gate.  We don’t consider any first come first serve, because flights has different timings of departure where other flights should wait.

DEFINITIONS IN GRAPH THEORY Graph coloring: Given a graph, color all the vertices so that two adjacent vertices get different colors. Complete Graphs: Every vertices should have an edge between each other vertices. Chromatic number: The chromatic number of a graph is the smallest number of colors needed to color the vertices, so that no two adjacent vertices share the same color the smallest value of possible to obtain a k-coloring.

PROPERTIES TO SOLVE GRAPH PROBLEM MAXIMUM DEGREE ORDERING: Chromatic number: Consider every vertex is of degree at most d. V 1)Then how many colors we need to color the graph?  For an uncolored vertex v, it has maximum d neighbors and thus at most d different colors. So, if we have d+1 colors, then we can always color it, by choosing a color not in its neighbors. V  Note: This is just a sufficient condition, but not necessary. For example, a tree could have large maximum degree, Vertex(v) but we can color it using only two colors.

PROPERTIES TO SOLVE GRAPH PROBLEM Claim: Suppose there is an ordering of the vertices v 1, …, v n, such that each vertex has at most d neighbors in front. Then the graph can be colored by d+1 colors. Proof. We color the vertices one by one following the ordering. When we color vertex v i, Maximum d neighbors of v i are colored. Since we have d+1 colors, we can always color v i using a color that has not appeared in its neighbors. We can repeat this process until every vertex is colored.

PROPERTIES TO SOLVE GRAPH PROBLEM INTERVAL GRAPH : It is the Intersection of intervals on the real line. It has one vertex for each interval in the family, and an edge between every pair of vertices corresponding to intervals that intersect.  In an interval graph G, there is a vertex with degree at most k-1. Let ω(G) = k Let v be the interval with leftmost right endpoint (earliest end point).  Any interval that intersects v must intersect v at the right endpoint, as otherwise v is not the interval wit leftmost right endpoint. V

PROPERTIES TO SOLVE GRAPH PROBLEM  So, all the intervals that intersect v must intersect with each other, and thus they form a complete subgraph. Since ω(G) = k, this complete subgraph is of size at most k, and thus v has at most k-1 neighbors.  Therefore, v is a vertex of degree at most k-1.

I.S TEPS TO SOLVE We can see timings of EIGHT flights in this chart. H G F E D C B A  This is before use of graph construction

I.S TEPS TO SOLVE Keep this time intervals in an Interval graph where we may see time conflicts between the flight timings. h g f e d c b a We can see in blue line 3 flights to land in airport at same time.

I.STEPS TO SOLVE Construct a Graph by using interval graph and arrange the vertices and edges. h a d c e f b g

I.S TEPS TO SOLVE  By using Graph Coloring we have colored following vertices and got different colors as follows. h a d c c e f b g

I.STEPS TO SOLVE By using chromatic color we have colored following vertices and got minimum K-colors. h a c d e f b 4-colors g 4-gates(red,yellow,blue,green)

I.P ORTRAY OF SOLUTION Colors which we got by edges are the gates which we are Allocating Gates For Flights. Finally the allocation of gates by using coloring graphs results to: Gates: Red Yellow Blue Green A,B F,D C,G H,E Suddenly if any new flight would like to land in airport then we can allot any gate by considering(check the time intervals of flights in all gates) that which runway is free at that time..

SUMMARY Graph coloring is an important problem in graph theory. It is useful in modeling problems in real life. And we can find an optimal coloring in some special cases. With this we can allocate timings for many applications like allocating gates for flights without time conflict.

REFERENCE: Example to explain aircraft problem _27#page-2http://link.springer.com/chapter/ %2F _27#page-2 Graph coloring Graphcoloring.(ppt) erial/DISCRETE%20STRUCTURES/2011/Graph %20Coloring.ppt erial/DISCRETE%20STRUCTURES/2011/Graph %20Coloring.ppt 4-dec11-f08.ppt 09.ppt

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