In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The k-planar crossing number of a graph cr k ( G) is the number of crossings required when every edge of G must be drawn in one of k distinct planes. It was shown in [2] that cr 2 ( Q 8 ) ≤ 256 which we improve to cr 2 ( Q 8 ) ≤ 128. Our approach highlights the relationship between symmetric drawings and the study of k-planar crossing numbers. We conclude with several open questions concerning this relationship

Clark, Gregory J.

Spencer, Gwen

English

Smith College: Smith ScholarWorks

Masthead Logo Smith ScholarWorksMathematics and Statistics: Faculty Publications Mathematics and Statistics6-2017How Low Can You Go? New Bounds on theBiplanar Crossing Number of Low-dimensionalHypercubesGregory J. ClarkUniversity of South Carolina - ColumbiaGwen SpencerSmith College, gspencer@smith.eduFollow this and additional works at: https://scholarworks.smith.edu/mth_facpubsPart of the Mathematics CommonsThis Article has been accepted for inclusion in Mathematics and Statistics: Faculty Publications by an authorized administrator of Smith ScholarWorks.For more information, please contact scholarworks@smith.eduRecommended CitationClark, Gregory J. and Spencer, Gwen, "How Low Can You Go? New Bounds on the Biplanar Crossing Number of Low-dimensionalHypercubes" (2017). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.https://scholarworks.smith.edu/mth_facpubs/39How Low Can You Go?New Bounds on the Biplanar Crossing Numberof Low-dimensional HypercubesGregory J. Clark∗, Gwen Spencer†June 2017AbstractIn this note we provide an improved upper bound on the biplanarcrossing number of the 8-dimensional hypercube. The k-planar crossingnumber of a graph crk(G) is the number of crossings required when everyedge of G must be drawn in one of k distinct planes. It was shown in [2]that cr2(Q8) ≤ 256 which we improve to cr2(Q8) ≤ 128. Our approachhighlights the relationship between symmetric drawings and the studyof k-planar crossing numbers. We conclude with several open questionsconcerning this relationship.1 IntroductionThe traditional crossing number of a graph G = (V,E), denoted by cr(G), isthe minimum number of edge crossings required to draw G in the 2-dimensionalEuclidean plane. To study printed circuit boards, Owens [4] generalized thequestion: what is the minimum number of edge crossings required by a draw-ing that is allowed to carefully divide the edges of G among two different 2-dimensional Euclidean planes? Since then the definition has been extended tok ≥ 2 planes [2].Suppose that E is partitioned into k disjoint subsets, E1, E2, ..., Ek, and letGi = (V,Ei). Each Gi has some crossing number cr(Gi). Suppose further thatGi will be drawn in the ith plane from a set of k distinct planes. The k−planarcrossing number of G, denoted crk(G) is then the minimum ofcr(G1) + cr(G2) + ...+ cr(Gk)∗University of South Carolina. Columbia, SC, USA.†Smith College. Northampton, MA, USA. This material is based upon work supportedby the National Science Foundation under Grant Number DMS 1641020 while both authorsattended the “Beyond Planarity: Crossing Numbers of Graphs” workshop (organized by theAmerican Mathematical Society), and under Grant No. DMS-1440140 while G. Spencer wasin residence at the Mathematical Sciences Research Institute in Berkeley, California, duringthe Fall 2017 semester.1arXiv:1711.01194v1  [math.CO]  3 Nov 2017over all partitions of the edge set E.Trivially, letting E1 = E shows that crk(G) ≤ cr(G). The question remains:given the freedom to consider any partition of G’s edges among k disjoint planes,how low can we drive the number of required crossings?A significant challenge in designing a crossing-minimizing k-planar drawingof G is that, even for quite simple Gi, cr(Gi) could be unknown. For example:for Q4, the 4-dimensional hypercube, it is known that cr(Q4) = 8; however, theexact value of cr(Qd) is unknown for d > 4 [3].The previous upper bound cr2(Q8) ≤ 256 was given by a construction ofCzabarka, Sy´kora, Sze´kely, and Vrt´o in [2]. Czabarka et al. give a generalconstruction for an upper bound on cr2(Qd) that achieves 256 crossings whend = 8. Their approach specifies a bi-planar partition of the edges of Q8 basedon a set of lower-dimensional hypercube subgraphs. Their upper bound is min-imized when these hypercube subgraphs are as-uniform-as-possible in size. Inparticular, for Q8 their construction specifies sixteen disjoint Q4 subgraphs inPlane 1 and a further sixteen disjoint Q4 subgraphs in Plane 2. Recall thatcr(Q4) = 8, so drawing each disjoint copy of Q4 optimally yieldscr2(Q8) ≤ 16× 2× 8 = 256.We now present our main result which improves on the the best known upperbound of cr(Q8) by a factor of 2.Theorem 1 There exists a 2-planar drawing of the 8-dimensional hypercubewith 128 crossings so that cr2(Q8) ≤ 128.2 A biplanar drawing of Q8 with 128 crossingsTo prove Theorem 1, we provide a biplanar drawing of Q8 with 128 crossings.We improve the previous construction by plane-swapping edges to give a netreduction in total edge crossings. Our drawing consists of graphs G1 and G2 inPlane 1 and 2 respectively such that G1 ∼= G2 where cr(Gi) ≤ 64. We foundseveral distinct bi-planar drawings of Q8 with exactly 128 crossings which satisfythese conditions. For ease of exposition, we present a highly symmetric drawing.We define a depleted n-dimensional hypercube to be a graph whose vertexset is V (Qn) and will refer to such graphs as depleted n-cubes. We will make useof depleted 5-cubes. To this end we introduce the following partition V (Q4) :=C1 unionsq C2 whereC1 := {0000, 1000, 0010, 1010, 0011, 1011, 0001, 1001}C2 := {0111, 1111, 0101, 1101, 0100, 1100, 0110, 1110}.For ease of notation, we denote cˆ ∈ C1 and cˇ ∈ C2. Moreover, we letb ∈ {0, 1} represent the usual binary-bit. Maintaining the notation of [2] werefer to each node of Q8 by a length-8 binary string from {0, 1}8. Given twobinary strings s1 and s2 we write s1s2, or s1− s2 for readability, to be the usualstring concatenation.2In our construction, each plane contains 512 edges, and furthermore, G1and G2 are isomorphic. For exposition, suppose that we initially have a Plane0 which contains all the edges and vertices of Q8. Further suppose that thereexist Planes 1 and 2 which each initially contain the vertices of Q8 and no edges.We move every edge from Plane 0 to either Plane 1 or Plane 2 to create ourbiplanar partition. In the following table, we describe explicitly the 512 edgeswe add to Plane 1.Consider the set of pairsP1 := {(0000, 1000), (0010, 1010), (0011, 1011), (0001, 1001)} ⊂(C12).For (cˆ1, cˆ2) ∈ P1 define the depleted 5-cube of Type 1, denoted D1(cˆ1, cˆ2), ac-cording to Table 1.E(D1(cˆ1, cˆ2)) for cˆ ∈ (cˆ1, cˆ2) ∈ P1(cˆ− b000, cˆ− b001) (cˆ− b000, cˆ− b100) (cˆ− b100, cˆ− b101) (cˆ− b001, cˆ− b101)(cˆ− b010, cˆ− b011) (cˆ− b010, cˆ− b110) (cˆ− b110, cˆ− b111) (cˆ− b011, cˆ− b111)(cˆ− b000, cˆ− b010) (cˆ− b001, cˆ− b011) (cˆ− b100, cˆ− b110) (cˆ− b101, cˆ− b111)(cˆ− 0101, cˆ− 1101) (cˆ− 0111, cˆ− 1111) (cˆ− 0110, cˆ− 1110) (cˆ− 0100, cˆ− 1100)(cˆ1 − 0000, cˆ2 − 0000) (cˆ1 − 0100, cˆ2 − 0100) (cˆ1 − 1100, cˆ2 − 1100) (cˆ1 − 1000, cˆ2 − 1000)(cˆ1 − 1001, cˆ2 − 1001) (cˆ1 − 1101, cˆ2 − 1101) (cˆ1 − 0101, cˆ2 − 0101) (cˆ1 − 0001, cˆ2 − 0001)Table 1: Table of the 64 edges of depleted 5-cubes of Type 1.The four depleted 5-cubes of Type 1 are vertex disjoint (from the form ofpairs in P1). We present an eight-crossing drawing of a depleted 5-cube of Type1 in Figure 2, which proves the following claim.Claim 1 cr(D1(cˆ1, cˆ2)) ≤ 8.We similarly define D2(cˇ1, cˇ2), the Depleted 5-cube of Type 2, according to Table2 givenP2 := {(0111, 1111), (0101, 1101), (0100, 1100), (0110, 1110)} ⊂(C22).Again, the four depleted 5-cubes of Type 2 are vertex disjoint. An eight-crossing drawing of a depleted 5-cube of Type 2 is given in Figure 2, whichproves the following claim.Claim 2 cr(D2(cˇ1, cˇ2)) ≤ 8.Each depleted 5-cube has 64 edges, so Plane 1 contains 512 edges. Further,no depleted 5-cube of Type 1 shares a vertex with a depleted 5-cube of Type 2.This follows from the form of the pairs in P1 and P2 and the form of the edgesets described in Tables 1 and 2. Thus, these 512 edges can be drawn in Plane1 with at most 64 crossings.3j − 0111j − 0110j − 0011j − 0010j − 0001j − 0101j − 0100j − 0000 j − 1000j − 1100j − 1101j − 1111j − 1110j − 1010j − 1011j − 1001k − 0000k − 0100k − 1100k − 1000k − 1010k − 1110k − 1111k − 1011k − 1001k − 1101k − 0101k − 0010k − 0110k − 0111k − 0011k − 0001Figure 1: A drawing of D1(cˆ1, cˆ2) for (cˆ1, cˆ2) ∈ P1 with eight crossings.Remark 1 Plane 2 contains all the edges of Q8 which are not in Plane 1.Moreover, G1 ∼= G2.We now provide a more illuminating description of the edges of Plane 2. Theedges in Plane 2 have a symmetric representation in terms of the edges in Plane1. Let ρ : E(Q8)→ E(Q8) such thatρ((vpvs, upus)) = (vsvp, usup)4E(D2(cˇ1, cˇ2)) for cˇ ∈ (cˇ1, cˇ2) ∈ P2.(cˇ− b000, cˇ− b001) (cˇ− b000, cˇ− b100) (cˇ− b100, cˇ− b101) (cˇ− b001, cˇ− b101)(cˇ− b010, cˇ− b011) (cˇ− b010, cˇ− b110) (cˇ− b110, cˇ− b111) (cˇ− b011, cˇ− b111)(cˇ− b000, cˇ− b010) (cˇ− b001, cˇ− b011) (cˇ− b100, cˇ− b110) (cˇ− b101, cˇ− b111)(cˇ− 0011, cˇ− 1011) (cˇ− 0001, cˇ− 1001) (cˇ− 0000, cˇ− 1000) (cˇ− 0010, cˇ− 1010)(cˇ1 − 0110, cˇ2 − 0110) (cˇ1 − 0111, cˇ2 − 0111) (cˇ1 − 0011, cˇ2 − 0011) (cˇ1 − 1011, cˇ2 − 1011)(cˇ1 − 1111, cˇ2 − 1111) (cˇ1 − 1110, cˇ2 − 1110) (cˇ1 − 1010, cˇ2 − 1010) (cˇ1 − 0010, cˇ2 − 0010)Table 2: Table of 64 edges of depleted 5-cubes of Type 2.where vp is a prefix string of length four, v1v2v3v4, and vs is a suffix stringof length four, v5v6v7v8 that together define vertex v = v1v2 . . . v8. Indeed ρcaptures the symmetric relationship between edges in Plane 1 and the edges inPlane 2. Assuming an ordering on the vertices of Q8 one can check that ρ isindeed a bijection. As an example, in Table 1 we assign edge (cˆb-000, cˆb-001)to Plane 1. So we sendρ((cˆb− 000, cˆb− 001)) = (b000− cˆ, b001− cˆ)to Plane 2. If we let Pi be the set of edges partitioned into Plane i thenP2 = ρ(P1). Moreover, the drawings provided in Figures 1 and 2 for depleted5-cubes of Type 1 (or Type 2, resp.) are also drawings of their images under ρ.It follows that, for the edge partition we describe, each plane can be drawn withat most 64 crossings implying that cr2(Q8) ≤ 128 as desired.A natural next step in this research is to determine whether or not thisbound is sharp. The authors believe this to be the case; however, such a proofremains elusive. Alas, we leave the reader with the following conjecture.Conjecture 1 cr2(Q8) = 128.3 Lower Bounds on structurally-symmetric k-planar crossing numbers for HypercubesNotably, our bi-planar drawing of Q8 satisfies G1 ∼= G2. This is a rather specialproperty and is termed self-complementary in [2]. It could be the case thatthere exists a non-isomorphic partition of E(Q8) which admits strictly fewercrossings. Yet, we wonder whether demanding that the Gi be isomorphic trulyforces a suboptimal number of crossings for k-planar drawings. In particular,such symmetry would be expected when considering highly symmetric graphslike hyper-cubes.To formalize this question, we introduce the following generalization of self-complementary edge partitions.Definition 1 For a finite graph G = (V,E), let P denote an edge-partition E =(E1, E2, ..., Ek) and define Gi = (V,Ei) for all i. If for all pairs (r, s) ∈ [k]× [k]we have Gr ∼= Gs, then P is a k-structurally-symmetric partition of G.5j − 0001j − 0000j − 0101j − 0100j − 0111j − 0011j − 0010j − 0110j − 1010j − 1110j − 1100j − 1000j − 1001j − 1101j − 1111j − 1011k − 0110k − 0010k − 1010k − 1110k − 1100k − 1000k − 1001k − 1101k − 1111k − 1011k − 0011k − 0100k − 0000k − 0001k − 0101k − 0111Figure 2: A drawing of D2(cˇ1, cˇ2) for (cˇ1, cˇ2) ∈ P2 with eight crossings.Trivially, when |E| is not a multiple of k, no k-structurally-symmetric parti-tion of E exists.Definition 2 If there exists a k-structurally-symmetric partition for G that canbe drawn with crk(G) crossings then we say that the graph G is k-structurally-symmetric.It is unclear whether graphs exist for which any k-structurally-symmetric6partition of E forces a sub-optimal k-planar drawing (which requires strictlymore than crk(G) crossings).In particular, we leave the reader with the following question.Question 1 Is the d-dimensional hypercube 2-structurally-symmetric?This question motivates the following definition.Definition 3 Let crkss(G) denote the minimum number of crossings requiredamong all k-structurally symmetric partitions of G. We call crkss the k-structurally-symmetric crossing number of G.Trivially, crkss(G) ≥ crk(G). So, k-structurally symmetric graphs are pre-cisely those graphs G that have crk(G) = crkss(G). We conclude by presentingthe reader questions concerning k-structurally-symmetric crossing numbers.Question 2 Characterize the set of all k-structurally-symmetric graphs. Tothis end, what structural properties ensure that a graph is k-structurally-symmtericor otherwise?Question 3 Provide a graph for which the difference between crkss(G) andcrk(G) is large (or even > 0). Further, is there an infinite family (Gn)n≥1such that Gn ⊆ Gn+1 and (crkss(Gn)− crk(Gn))n≥1 ↑ ∞?4 AcknowledgementsThis material is based upon work that started at the Mathematics ResearchCommunities workshop “Beyond Planarity: Crossing Numbers of Graphs”, or-ganized by the American Mathematical Society, with the support of the NationalScience Foundation under Grant Number DMS 1641020.We would like to extend our thanks to the organizers of the workshop fortheir commitment to engendering academic growth in young career mathemati-cians. We are particularly thankful for La´szlo´ Sze´kely and his exemplary men-toring which made this project possible.References[1] A. Aggarwal, M. Klawe, P. Shor, Multi-layer grid embeddings for VLSI,Algorithmica 6 (1991) 129-151.[2] E´. Czabarka, O. Sy´kora, L.A. Sze´kely, I. Vrt´o, Biplanar crossing numbers:a survey of results and problems, in: Gyo¨ri, E., Katona, G.O.H., Lova´sz,L. (Eds.), More Sets, Graphs and Numbers, Bolyai Society MathematicalStudies, vol. 15, Springer, Berlin, 2006, pp.57-77.[3] Faria, L., de Figueiredo, C. M. H., On the Eggleton and Guy conjecturedupper bound for the crossing number of the n-cube, Mathematica Slovaca50 (2000), 271-287.7[4] A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory18 (1971) 277-280.[5] J. Pach, L. A. Sze´kely, Cs. D. To´th, G. To´th, Note on k-planar crossingnumbers, to appear in Computational Geometry: Theory and ApplicationsSpecial Issue in Memoriam Ferran Hurtado.[6] F. Shahrokhi, O. Sy´kora, L. A. Sze´kely and I. Vrt´o, k-planar crossing num-bers, Discrete Appl. Math. 155 (2007), 1106-1115.8

How Low Can You Go? New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes

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How Low Can You Go? New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes

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