Fractional and Circular Separation Dimension of Graphs

The separation dimension of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ 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 $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt 2$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs and $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$.