GRAPH COLORING AND CLASSIFYING TROPICAL FISH By Vennam Chandrasekhar Reddy

A GENDA Problem statement Graph Construction Relation to graph problem Special Property Problem solution Comments References

P ROBLEM D EFINITION : A tropical fish hobbist had six different types of fish: Alphas, Betas, Certas, Deltas, Epsalas and Fetas which are designated by A, B, C, D, E, and F respectively. Because of predator-prey relationships, water conditions, and size only some types of fishes can survive with some other types of fishes in the same tank.

P ROBLEM D EFINITION : TypeABCDEF Cannot be with B, CA,CA, B, D, EB,C C, F E The following table gives information about the fishes that cannot be together : Our task is to arrange the fishes in a minimum number of Tanks.

G RAPH C ONSTRUCTION To model the situation, we simply need to construct a graph in which each vertex represents one of the types of fish and each edge connects vertices that are not compatible. The graph thus constructed turns out to be an interval graph.

G RAPH C ONSTRUCTION Vertex : Fish type Edge: Not Compatible A B C D E F

R ELATION TO A GRAPH PROBLEM Set of interval’s D F BE A C

R ELATION TO A GRAPH PROBLEM This Real world problem is converted to “interval graph coloring problem”. An “interval graph” is the graph showing intersecting intervals on a line. So, we associate a set of intervals I={I1,…,In} on a line with the interval graph G=(V,E),where V={1,…,n} and two vertices, x and y, are linked by an edge if and only if Ix∩Iy≠.

R ELATION TO A GRAPH PROBLEM E ( G ) = {{ v i, v j } | I X ∩ I Y ≠ ∅ } From the graph, it is an interval graph which is an undirected graph formed from a set of intervals I(1,2….10).

S PECIAL PROPERTY Umbrella Free Ordering: For every interval graph there will be an Umbrella Free-Ordering it states that, arranging the vertices in an order such that if there is an edge between two vertices then any edge that lies between the two vertices must be adjacent to the right vertex in the ordering.

S PECIAL PROPERTY An umbrella-free representation of a graph G is a concatenation (in any order) of all its connected component umbrella-free representations. UF be an umbrella-free representation of G, the vertices of two distinct connected components are not interleaved in UF.

S PECIAL PROPERTY So ordering of our graph using umbrella-free property is:A,B,D,C,E,F UMBRELLA FREE-ORDERING

P ROBLEM SOLUTION In this Umbrella Free-Ordering, we place the colours in a certain order. We can now color the vertices. We start by assigning blue to F, then Red to E.

P ROBLEM SOLUTION The C vertex, which is the next to be colored, can be colored in blue or other Color because Blue is already available color, we choose it for vertex C. We continue in this way until we colored the whole graph.

COLORING USING THE UMBRELLA FREE-ORDERING D 2 F 1 B 3E 2 A 2 C 1

P ROBLEM S OLUTION Assigning fishes to tanks, where the compatibility between fishes is considered. This problem comes down to colouring the vertices of an interval graph under the constraint that two vertices linked by an edge cannot be of the same colour.

P ROBLEM SOLUTION Vertex : Fish type Edge: Not Compatible A A B B C C D D E E F F

P ROBLEM SOLUTION This coloring requires Three colors, which means we need three tanks to assign 6 types of fishes. Blue TankRed TankBlack Tank Fetas & CertasAlphas, Deltas & EpsalasBetas

PROBLEM SOLUTION A A B B C C D D E E F F

C OMMENTS If this is not the case i.e: if the fish combinations are different interval graph is not constructed an arbitrary graph is constructed For arbitrary graphs no body knows correct algorithms to colour those arbitrary graphs.

R EFERENCES en/documents/NotesChap3.pdf X

R EFERENCES THANK YOU ANY QUERIES