Fractional and circular separation dimension of graphs

Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let π∘(G) be the number of circular orderings needed to separate all pairs and πf∘(G) be the fractional version. Among our results: (1) π∘(G)=1 if and only G is outerplanar. (2) π∘(G)≤2 when G is bipartite. (3) π∘(Kn)≥log2log3(n−1). (4) πf∘(G)≤32 for every graph G, with equality if and only if K4⊆G. (5) πf∘(Km,m)=3m−32m−1.