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A Bipartite Strengthening of the Crossing Lemma
"... Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any ..."
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Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. We prove for every k ∈ N that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least ckm 2 /n 2, such that every edge in E1 crosses all edges in E2, where ck> 0 only depends on k. This bound is best possible up to the constant ck for every k ∈ N. We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with xmonotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m 2 /(n 2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct xmonotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m 2 /(n 2 log(m/n))). 1
DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
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The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1