Upper Bounds on Sets of Orthogonal Colorings of Graphs

Details Upper Bounds on Sets of Orthogonal Colorings of Graphs Upper Bounds on Sets of Orthogonal Colorings of Graphs

2011-10-11

arxiv.org/abs/1110.2237v2

Mathematics

Combinatorics

17 pages

Scientific paper

We generalize the notion of orthogonal latin squares to colorings of simple graphs. We define two $n$-colorings of a graph to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs. We also show that the problem of finding a transversal in a latin square of order $n$ is equivalent to finding an $n$-clique in a particular graph.

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