Children disliking in square $n^2\geq 4$ children are to be placed in an $n\times n$ square. Some pairs of children dislike each other and do not want to be next to each other. Disliking is mutual, and being next to each other means being directly above/below/left/right of each other. What is the maximum $k$ such that if each child dislikes no more than $k$ other children, then a placement is always possible?
If a child dislikes $n^2-2$ other children, a placement is obviously not possible because we cannot find two children to be next to this child even if we place the child in a corner. To do better than this, we can divide the children into two groups of roughly equal size, and let each child dislike all children in the other group. Then each child dislikes about $n^2/2$ other children, and clearly we cannot place them in the square in the desired way.
 A: (Only an initial entry. Not an actual solution)
Just for starters we can at least show that k=2 must be a lower bound for n>2 even though that was probably pretty expected.
If the children dislike at most 2 other children then the children can be grouped into disjoint "chains of dislike", "cycles of dislike", or "lone children".
Essentially the connected components of the graph where edges denote dislike are either lone vertices, chains or cycles.
In that case the following scheme can be adopted (in the case where n is odd)


*

*List the connected components

*Enumerate the children in each component in the natural way and rolling over to the next component. 

*Place child 1 in the upper left corner

*Place the next child two steps to the right of the previous child and roll over to next row if necessary. (And repeat)


If n is even replace 4 with


*If n is even place the next child three steps to the right of the previous child and roll over to next row if necessary. (And repeat)


Implementing the algorithm for an n=3 case runs as:

What at least might be a good next step would perhaps be and investigate whether k=3 seems to not have seating solutions for some n and thus move towards determining whether or not the maximal k is general or depends on n. 
