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Hao, Hanson

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Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyOn a Computer-Aided Decomposition of theComplete Digraph into Orientations of K4 − ewith a Double EdgeHanson Hao, Claudia ZhuIllinois Mathematics and Science AcademyMentor: Dr. Saad El-Zanati (Illinois State University)International Student Science Fair, IMSA, June 30, 20181 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyGraphs and Directed Graphs• Graphs are objects comprising ofa vertex set and an edge set,where the edges connect thevertices.• In a directed graph, or digraph,the edges (called arcs) areassigned directions and aretreated as ordered pairs.• Each different way to assigndirections to the edges is calledan orientation.• The complete digraph on nvertices, denoted K∗n, is thedigraph with the arcs (a, b) and(b, a) between every pair ofvertices a and b.• The number of arcs in K∗n isn(n− 1).0120120122 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyDigraph Decompositions• If D and H are digraphs with D a subgraph of H, then aD-decomposition of H is a partition of the arc set of H intosubgraphs isomorphic to D, called D−blocks.• The spectrum for a digraph D is the set of all n for which aD-decomposition of K∗n exists.• Let V (K∗n) = Zn and let D be a subgraph of K∗n. Definerotating D or clicking D as the isomorphism i 7→ i+ 1 foreach vertex in V (D).• A D-decomposition of K∗n is cyclic if clicking D preserves theD-blocks of the decomposition.3 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographySome Background and Previous ResultsHartman and Mendelsohn (1986) found the spectra for allsubgraphs of K∗3 .11A. Hartman and E. Mendelsohn, The Last of the Triple Systems, Ars Combin. 22 (1986), 25-41.4 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographySome Background and Previous ResultsBunge et al. (2017)– submitted found spectra for almost all of thebelow 10 orientations of K4 − e.2wxyzwxyzwxyzwxyzwxyzwxyzwxyzwxyzwxyzwxyz2R. C. Bunge, B. D. Darrow Jr., T. M. Dubczuk, Mentor, Competition Entrant, G. L. Keller, G. A. Newkirk,and D. P. Roberts, On Decomposing the Complete Symmetric Digraph into Orientations of K4-e,Discussiones Mathematicae-Graph Theory, submitted.5 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyOur Research Question• We want to find the spectrum for all D such that D is anorientation of K4 − e with a double edge.wxyz• When the graph is oriented, one of the two double edges fromw to y must be directed towards w and the other one must bedirected towards y.• The digraphs are named using the conventions in An Atlas ofGraphs by Read and Wilson.6 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographySome Useful ObservationsThe orientation of a digraph D may be reversed by changing thedirection of the arrow on each arc. We denote the reverseorientation as Rev(D).Observe that if D and H are digraphs, then a D-decomposition ofH exists if and only if a Rev(D)-decomposition of Rev(H) exists.Since K∗n ∼= Rev(K∗n), we have:ObservationIf D is a digraph, then D decomposes K∗n if and only if Rev(D)decomposes K∗n.7 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyOur DigraphsAfter writing out the 16 possible orientations and taking out anydigraphs that were isomorphic and/or reverses of each other, weobtained 5 digraphs of interest:D125[w, x, y, z]wxyzD110[w, x, y, z]wxyzD106[w, x, y, z]wxyzD118[w, x, y, z]wxyzD135[w, x, y, z]wxyz8 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyNecessary Conditions for Digraph DesignsThe following are the necessary conditions for a D-decompositionof K∗n to exist.Order condition: |V (D)| ≤ n.Size condition: |E(D)| divides n(n− 1).Degree condition: Both gcd{outdegree(v) : v ∈ V (D)} andgcd{indegree(v) : v ∈ V (D)} divide n− 1, which isboth the indegree and outdegree of every vertex inK∗n.• Need 6|n(n− 1) by condition 2. Thus n ≡ 0, 1, 3, or 4mod 6.• Need n ≥ 4 by condition 1.9 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD110-Decomposition of K∗6We now demonstrate some designs for small n which use theaforementioned clicking mechanism.wxyz∞0123410 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD110-Decomposition of K∗6We now demonstrate some designs for small n which use theaforementioned clicking mechanism.wxyz∞0123410 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD110-Decomposition of K∗6We now demonstrate some designs for small n which use theaforementioned clicking mechanism.wxyz∞0123410 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD110-Decomposition of K∗6We now demonstrate some designs for small n which use theaforementioned clicking mechanism.wxyz∞0123410 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD110-Decomposition of K∗6We now demonstrate some designs for small n which use theaforementioned clicking mechanism.wxyz∞0123410 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyD135-Decomposition of K∗7wxyz012345611 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyMain Idea• Not all decompositions need to be cyclic; some can bemanual. Finding these manual decompositions was aided by acomputer.• A digraph D may be considered numerically as a set ofordered pairs (Indegree, Outdegree) for each vertex.• K∗6 may be represented as{(5, 5), (5, 5), (5, 5), (5, 5), (5, 5), (5, 5)}.• D135, which is below, may be represented as[w, x, y, z]→ {(2, 2), (1, 1), (2, 2), (1, 1)}.wxyz12 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyMain Idea• Decompositions can be considered similarly by "adding" theset of ordered pairs corresponding to each D−block to createthe graph we wish to decompose.• The set can be "permuted" in any way.• To decompose K∗6 with D135, start with{(0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0)}.• Permute the set: w → 5, x→ 3, y → 6, z → 4.• Add the permutation to the graph, which then becomes{(0, 0), (0, 0), (1, 1), (1, 1), (2, 2), (2, 2)}.13 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyManual D135-Decomposition of K∗6• {(0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0)}• {(0, 0), (0, 0), (1, 1), (1, 1), (2, 2), (2, 2)}• {(2, 2), (1, 1), (1, 1), (1, 1), (4, 4), (3, 3)}• {(3, 3), (1, 1), (3, 3), (1, 1), (5, 5), (5, 5)}• {(4, 4), (3, 3), (4, 4), (3, 3), (5, 5), (5, 5)}• {(5, 5), (5, 5), (5, 5), (5, 5), (5, 5), (5, 5)}14 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyCode Algorithm• The code does the above process backwards: it starts at theend product (i.e. {(5, 5), (5, 5), (5, 5), (5, 5), (5, 5), (5, 5)})and subtracts set permutations until we reach{(0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0)}.• The code also uses a memoization technique in order toreduce runtime.• In the memoization technique, the code stores previous failedruns so it does not have to recompute the same negativeresult again in the future.• Many important building blocks were found using the code.15 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyImpossibility Results• There does not exist a D106- decomposition of K∗6p or K∗6q+3for any p ≥ 1, q ≥ 1.• There does not exist a D118- decomposition of K∗6p or K∗6q+4for any p ≥ 1, q ≥ 1.16 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyImpossibility Results–Example Proof• There does not exist a D118- decomposition of K∗6p or K∗6q+4for any p ≥ 1, q ≥ 1.• Apply necessary condition C, which is:• Both gcd{outdegree(v) : v ∈ V (D)} andgcd{indegree(v) : v ∈ V (D)} divide n− 1.• gcd{outdegree(v) : v ∈ V (D118)} = gcd(2, 2, 2, 0) = 2. Bycondition C, if a D118- decomposition of K∗n existed, then wemust have 2|(n− 1). Thus, (n− 1) ≡ 0 mod 2. However,(6p− 1) ≡ (6q + 3) ≡ 1 mod 2 6≡ 0 mod 2, so the necessarycondition fails and thus there does not exist a D118-decomposition of K∗6p or K∗6q+4 for any p ≥ 1, q ≥ 1.17 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyBlow-up Constructions• In building general constructions from small cases, we utilize thefollowing theorems from previous literature:3• If n is odd, then a {K3,K5}-decomposition of Kn exists.• The necessary and sufficient conditions for the existence of aK3-decomposition of Ku×m are(i) u ≥ 3,(ii) (u− 1)m ≡ 0 (mod 2), and(iii) u(u− 1)m2 ≡ 0 (mod 6).• If u ≥ 3 and u ≡ 0 (mod 3), then there exists a K3-decompositionof Ku×2,4.• Let m, r, s, t, u1, u2, . . . , um all be positive integers. If thereexists a {Kr,Ks}-decomposition of Ku1,u2,...,um , then there alsoexists a {Kr×t,Ks×t}-decomposition of Ktu1,tu2,...,tum .3C. J. Colbourn and J. H Dinitz (Editors), Handbook of Combinatorial Designs, 2nd ed.,Chapman & Hall/CRC Press, Boca Raton, FL, 2007.18 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyBlow-up Constructions• If n ≡ 0 mod 6 and n ≥ 6, then a (K∗n,D) design exists forD ∈ {D135}.• If n ≡ 1 mod 6 and n ≥ 7, then a (K∗n,D) design exists forD ∈ {D135}.19 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyConclusions and Future Work• We were able to make general constructions and impossibilityarguments for some of the cases.• The majority of cases produced partial results.• Our code could be optimized to reduce runtime and memoryusage, as both were impediments when trying to brute-forcethrough larger cases.20 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyAcknowledgements• We’d like to thank our mentor, Dr. Saad El-Zanati, withoutwhose guidance and support this project would have beenimpossible.21 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyR. J. R. Abel, F. E. Bennett, and M. Greig, “PBD-Closure,” inHandbook of Combinatorial Designs, C. J. Colbourn andJ. H. Dinitz (Editors), 2nd ed., Chapman & Hall/CRC Press,Boca Raton, FL, 2007, pp. 246–254.J.-C. Bermond, C. Huang, A. Rosa, and D. Sotteau,Decomposition of complete graphs into isomorphic subgraphswith five vertices, Ars Combin., 10 (1980), 211–254.J.-C. Bermond and J. Schönheim, G-decomposition of Kn,where G has four vertices or less, Discrete Math. 19 (1977),113–120.R. C. Bunge, B.D. Darrow Jr., T.M. Dubczuk, Mentor,Competition Entrant, G. L. Keller, G. A. Newkirk, andD. P. Roberts, On Decomposing the Complete SymmetricDigraph into Orientations of K4-e, DiscussionesMathematicae–Graph Theory, submitted.21 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyR. C. Bunge, Mentor, H. J. Fry, K. S. Krauss, D. P. Roberts,C. A. Sullivan, A. A. Unsicker, and N. E. Witt, On the Spectraof Bipartite Directed Subgraphs of K∗4 , J. Combin. Math.Combin. Comput., 98 (2016), 375–390.C. J. Colbourn and J. H. Dinitz (Editors), Handbook ofCombinatorial Designs, 2nd ed., Chapman & Hall/CRC Press,Boca Raton, FL, 2007.G. Ge, “Group divisible designs,” in Handbook ofCombinatorial Designs, C. J. Colbourn and J. H. Dinitz(Editors), 2nd ed., Chapman & Hall/CRC Press, Boca Raton,FL, 2007, pp. 255–260.G. Ge, S. Hu, E. Kolotoğlu, and H. Wei, A complete solutionto spectrum problem for five-vertex graphs with application totraffic grooming in optical networks, J. Combin. Des. 23(2015), 233–273.21 / 21Introduction Research Question Basic Results Computer-Generated Results General Results Conclusions BibliographyA. Hartman and E. Mendelsohn, The Last of the TripleSystems, Ars Combin. 22 (1986), 25–41.R. C. Read and R. J. Wilson, An Atlas of Graphs, OxfordUniversity Press, Oxford, 1998.21 / 21

A Computer-Aided Decomposition of the Complete Digraph into Orientations of K4 − e with a Double Edge

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A Computer-Aided Decomposition of the Complete Digraph into Orientations of K4 − e with a Double Edge

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