Fractional and circular separation dimension of graphs

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Abstract

The separation dimension of a graph G, written π(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 πf(G), which is the minimum of ab 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 πf(G)3, with equality if and only if G contains K4. There is no stronger bound even for bipartite graphs, since πf(Km,m)=πf(Km+1,m)=3mm+1. We also compute πf(G) for cycles and some complete tripartite graphs. We show that πf(G)<2 when G is a tree and present a sequence of trees on which the value tends to 43.

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(n1). (4) πf(G)32 for every graph G, with equality if and only if K4G. (5) πf(Km,m)=3m32m1.

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