Question

Bonus problem: Use graph theory to solve the following problem. Let A be an n by n array in which no two rows are identical. Show that it is possible to delete some column so that in the resulting n by n-1 array, no two rows are equal. . LetA rows are identical. Show that it impos

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Answer 1

First we create a matrix from a graph since the problem asks to be solved using graph theory.

Let $G$ be a graph on $n$

We create a matrix $M=(a_{ij})$ from $G$  as follows:

If there exists any edge between two vertices $i,j$ we write $a_{ij}=1$ otherwise we put it $0$

Thus we get a $n\times n$ matrix which we call $A$.

Now assume that there exists no such column $C$ of $A$ such that deleting $C$ will result in at least two rows being identical

which in turn implies that if we delete any column of $A$ then at least two rows will become same.

Thus it implies that the two rows differed only in the entries of that column.

But note that thuis is true for any column of A.

But in the matrix A we only have two distinct elements $0\& 1$.

Thus it is absurd because all the rows are given to be non-identical in $A$.

Thus there exists at least one column whose deletion results in a $n\times n-1$ array with no two rows same.