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Multicolored Cubes

Multicolored Cubes. Presented by K. LAKSHMI PRABHA, Assistant Professor, Department of Mathematics, Rajapalayam Rajus’ College, Rajapalayam. Abstract. I am going to present a paper on an Application of graph theory.

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Multicolored Cubes

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  1. Multicolored Cubes Presented byK. LAKSHMI PRABHA, Assistant Professor, Department of Mathematics,Rajapalayam Rajus’ College,Rajapalayam

  2. Abstract I am going to present a paper on an Application of graph theory. In this presentation we are going to solve a problem using graph theory concept.

  3. Problem • We are given four cubes. The six faces of every cube are variously colored blue, green, red, and white. Is it possible to stack the cubes one on top of another to form a column such that no color appears twice on any of the four sides of this column?

  4. Cubes Cube 2 Cube 1 B R W G R G R B W G W R Cube 3 Cube 4 W B R B R W G B W G G R

  5. Graph A graph is an ordered pair G = (V, E) where V is a non-empty set and E is the two element subset of unordered pair of elements of V.

  6. Degree • The degree of a point vi in a graph G is the number of lines incident with vi. The degree of vi is denoted by dG(vi) or deg vi or simply d(vi).

  7. Subgraph • A graph H = (V1, X1) is called a subgraph of G = (V, X) if V1 is the subset of V and X1 is the subset of X.

  8. Self Loop • An edge to be associated with a vertex pair (vi, vi) such an edge having the same vertex as both its end vertices is called a self-loop.

  9. Simple graph • A graph is simple if it has no loops and two of its edges join the same pair of vertices.

  10. Solution • Step 1: Draw a graph with four vertices B, G, R, and W . If a blue face in cube 1 has a white face opposite to it, draw an edge between vertices B and W in the graph. Do the same for the remaining two pairs of faces in cube 1.

  11. Put label 1 on all three edges resulting from cube 1. • Repeat the procedure for the other cubes one by one on the same graph.

  12. Graph 4 1 R B 3 1 3 2 1 2 4 2 W G 4 3

  13. Step 2: Consider the above graph. The degree of each vertex is the total number of faces with corresponding color. Consider two opposite vertical sides of the desired column of four cubes, say facing north and south.

  14. A subgraph will represent these eight faces – four facing south and four north. • Each of the four edges in this subgraph will have a different label – 1, 2, 3 and 4. • Moreover, no color occurs twice on either the north side or south side of the column if and only if every vertex in this subgrpah is of degree two.

  15. Subgraph 4 B R 1 2 W G 3

  16. Exactly the same argument applies to the other two sides, east and west, of the column. 3 R B 1 4 W G 2

  17. Thus the four cubes can be arranged if and only if there exist two edge – disjoint subgraphs, each with four edges, each of the labeled differently, and such that each vertex is of degree two.

  18. Thank You

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