Maximum Size of a Family of Pairwise Graph-Different Permutations

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.