The separation dimension of a graph , written , is the minimum number of linear orderings of such that every two nonincident edges are “separated” in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension
, which is the minimum of such that some linear orderings (repetition allowed) separate every two nonincident edges at least times.
In contrast to separation dimension, fractional separation dimension is bounded: always , with equality if and only if contains . There is no stronger bound even for bipartite graphs, since . We also compute for cycles and some complete tripartite graphs. We show that when is a tree and present a sequence of trees on which the value tends to .
Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let be the number of circular orderings needed to separate all pairs and be the fractional version. Among our results: (1) if and only is outerplanar. (2) when is bipartite. (3) . (4) for every graph , with equality if and only if . (5) .