$\text{(I)}{x}^2-19x+78=0$

$\text{(II)}{y}^2-11y+18=0$

The correct option is If x= y or no relation can be established.

To determine the relationship between $x$ and $y$, we need to solve both quadratic equations step-by-step.

Step 1: Solve equation (I) for $x$

The given equation is:

$x^2-19x+78=0$

We need to find two numbers that multiply to give $78$ and add up to give $-19$.
These two numbers are $-13$ and $-6$.

Splitting the middle term:

$x^2-13x-6x+78=0$

$x(x-13)-6(x-13)=0$

$(x-13)(x-6)=0$

Therefore, the values of $x$ are:

$x=13$ or $x=6$

Step 2: Solve equation (II) for $y$

The given equation is:

$y^2-11y+18=0$

We need to find two numbers that multiply to give $18$ and add up to give $-11$.
These two numbers are $-9$ and $-2$.

Splitting the middle term:

$y^2-9y-2y+18=0$

$y(y-9)-2(y-9)=0$

$(y-9)(y-2)=0$

Therefore, the values of $y$ are:

$y=9$ or $y=2$

Step 3: Compare the values of $x$ and $y$

Now, let us compare each value of $x$ with each value of $y$:

1. When $x=13$ and $y=9$, $x>y$
2. When $x=13$ and $y=2$, $x>y$
3. When $x=6$ and $y=9$, $x<y$
4. When $x=6$ and $y=2$, $x>y$

Since we obtain both $x>y$ and $x<y$ depending on the pair of roots, no consistent relationship can be established between $x$ and $y$.

Hence, the correct choice is If x= y or no relation can be established.