Apart from two possible exceptions, the design spectrum has been determined for every graph with six vertices and at most eight edges. The purpose of this note is to establish the existence of the two missing designs, both of order 32

Forbes, Anthony D.

Forbes, Kim A.

Griggs, Terry S.

English

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Open Research OnlineThe Open University’s repository of research publicationsand other research outputsCompleting the design spectra for graphs with sixvertices and eight edgesJournal ItemHow to cite:Forbes, Anthony D.; Griggs, Terry S. and Forbes, Kim A. (2018). Completing the design spectra for graphswith six vertices and eight edges. Australasian Journal of Combinatorics, 70(3) pp. 386–389.For guidance on citations see FAQs.c© [not recorded]Version: Version of RecordLink(s) to article on publisher’s website:https://ajc.maths.uq.edu.au/pdf/70/ajc v70 p386.pdfCopyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.oro.open.ac.ukAUSTRALASIAN JOURNAL OF COMBINATORICSVolume 70(3) (2018), Pages 386–389Completing the design spectra for graphs withsix vertices and eight edgesAnthony D. Forbes Terry S. GriggsSchool of Mathematics and StatisticsThe Open UniversityWalton Hall, Milton Keynes MK7 6AAU.K.anthony.d.forbes@gmail.com terry.griggs@open.ac.ukKim A. ForbesResearch, Business and InnovationKingston UniversityPenrhyn Road, Kingston upon Thames, KT1 2EEU.K.kimaforbes@gmail.comAbstractApart from two possible exceptions, the design spectrum has been de-termined for every graph with six vertices and at most eight edges. Thepurpose of this note is to establish the existence of the two missing de-signs, both of order 32.1 IntroductionLet G be a simple graph. If the edge set of a simple graph K can be partitioned intoedge sets of graphs each isomorphic to G, we say that there exists a decomposition ofK into G. In the case where K is the complete graph Kn we refer to the decompo-sition as a G-design of order n. The design spectrum of G is the set of non-negativeintegers n for which there exists a G-design of order n.The design spectrum problem for small graphs has attracted attention. In par-ticular, it has been solved for (i) all graphs with at most ﬁve vertices, (ii) all graphswith six vertices and at most seven edges, (iii) all graphs with six vertices and eightedges, with two possible exceptions, and (iv) eleven of the 21 graphs with six ver-tices and nine edges. See [1] and [2] for details and references. More recently, thespectrum problem was resolved for all of the remaining ten graphs with six verticesand nine edges, [4].A.D. FORBES ET AL. /AUSTRALAS. J. COMBIN. 70 (3) (2018), 386–389 387Figure 1: Graphs M1 and M2    c a eb d f    b a ed c fM1 M2Figure 2: Graph Q  c ab d2 Design constructionThe two possible exceptions for graphs with six vertices and eight edges are illustratedin Figure 1, and in each case it is order 32 that remains unresolved; see [5], where thegraphs are denoted by M1 and M2. To complete the task of determining the designspectra for 6-vertex, 8-edge graphs it suﬃces merely to exhibit designs of order 32for M1 and M2. However, it might be of some interest to explain how they wereobtained; so we give brief details.The number of graphs in each design is 62 and without a vertex of degree 1 thereis no convenient automorphism that we can exploit. So we are left in each case withthe task of assembling sixty-two 6-tuples of numbers from the set N = {0, 1, . . . , 31}.Let Q denote the graph obtained by removing an edge from a K4 (Figure 2). Ourprimary objective is to obtain a decomposition D of K32 labelled with N into 62triangles and 62 copies of Q, also labelled from N .A partial Steiner triple system PSTS(x) consists of a point set, X say, of cardi-nality x and a collection of triples, 3-element subsets of X, such that each unorderedpair of elements from X occurs in at most one triple. We create by hill climbing,[6] (or see [3, §2.7.2]), a random PSTS(32), S, with point set N and 62 triples. Thetriples of S are the triangles of D.Next, we create the T -K matrix M for assembling the graphs Q. Denote by Tthe set of edges of the labelled K32 that are not present as pairs in the triples of S.A.D. FORBES ET AL. /AUSTRALAS. J. COMBIN. 70 (3) (2018), 386–389 388To each ordered 4-tuple k = (a, b, c, d), a, b, c, d ∈ N , we associate a graph Qk ∼= Qwith its vertices labelled by the elements of k as shown in Figure 2. Let K be the setof ordered 4-tuples k corresponding to distinct labelled Qk such that all the edges ofQk belong to T . The rows of M are indexed by elements of T and the columns byelements of K. The matrix is deﬁned by Mt,k = 1 if Qk contains edge t; otherwiseMt,k = 0.We attempt to solve Mv = 1 for v a {0, 1} vector. If we are unsuccessful, we tryanother PSTS(32). Otherwise we recover the 4-tuples corresponding to 1s in v andhence the graphs Q of our decomposition D.Finally, for δ = 2, 3 we pair oﬀ triangles and Qs in D such that in each pairthere is a common label attached to the triangle and a vertex of degree δ in the Q.Thus we have the required decompositions of K32 into M1 when δ = 3 and M2 whenδ = 2. The results are presented in Section 3. Incidentally, by pairing triangles andQs with no common vertices we obtain a Q ∪K3 design of order 32.3 The designsWe represent the graphs by ordered 6-tuples (a, b, c, d, e, f)M1 and (a, b, c, d, e, f)M2,where the letters correspond to vertices as illustrated in Figure 1. With vertex setZ32 the decompositions consist of 62 graphs each:(8, 9, 3, 16, 23, 0)M1, (5, 8, 6, 27, 25, 1)M1, (20, 7, 0, 27, 1, 16)M1,(3, 19, 5, 20, 6, 0)M1, (14, 20, 5, 22, 1, 4)M1, (20, 21, 18, 31, 24, 2)M1,(8, 11, 14, 25, 19, 1)M1, (26, 17, 8, 19, 27, 13)M1, (30, 18, 8, 19, 0, 27)M1,(15, 11, 2, 27, 29, 0)M1, (14, 13, 0, 15, 27, 3)M1, (10, 14, 2, 19, 18, 4)M1,(14, 7, 16, 30, 26, 28)M1, (9, 18, 14, 26, 22, 0)M1, (23, 14, 17, 31, 3, 18)M1,(25, 21, 0, 14, 4, 6)M1, (0, 11, 19, 31, 1, 24)M1, (12, 10, 0, 16, 20, 26)M1,(26, 2, 0, 21, 1, 15)M1, (5, 18, 0, 12, 16, 2)M1, (19, 15, 6, 21, 27, 2)M1,(16, 13, 4, 19, 26, 3)M1, (22, 16, 15, 31, 27, 1)M1, (23, 16, 11, 28, 4, 20)M1,(24, 21, 3, 16, 4, 8)M1, (5, 23, 15, 26, 10, 30)M1, (28, 15, 3, 18, 0, 4)M1,(12, 24, 15, 27, 14, 6)M1, (4, 21, 5, 27, 15, 17)M1, (31, 27, 10, 25, 15, 30)M1,(17, 18, 2, 27, 20, 10)M1, (12, 4, 9, 31, 21, 7)M1, (31, 7, 3, 26, 2, 9)M1,(6, 18, 7, 31, 9, 1)M1, (22, 18, 13, 24, 5, 11)M1, (13, 31, 5, 28, 20, 8)M1,(6, 2, 13, 28, 16, 30)M1, (18, 1, 11, 29, 25, 16)M1, (6, 24, 10, 26, 20, 29)M1,(22, 25, 7, 26, 8, 10)M1, (26, 4, 11, 30, 29, 10)M1, (19, 25, 12, 28, 23, 9)M1,(29, 22, 4, 19, 31, 8)M1, (2, 3, 4, 29, 8, 12)M1, (23, 25, 2, 29, 27, 6)M1,(21, 22, 6, 23, 28, 8)M1, (30, 22, 2, 28, 12, 23)M1, (12, 22, 3, 17, 13, 11)M1,(28, 12, 1, 29, 9, 27)M1, (7, 1, 2, 13, 8, 15)M1, (10, 23, 1, 13, 15, 25)M1,(30, 1, 3, 21, 13, 24)M1, (11, 10, 3, 21, 20, 28)M1, (9, 17, 5, 21, 15, 20)M1,(29, 7, 5, 17, 13, 21)M1, (24, 28, 5, 17, 29, 14)M1, (7, 10, 9, 28, 19, 4)M1,(24, 7, 11, 23, 31, 19)M1, (17, 11, 6, 30, 0, 16)M1, (25, 9, 13, 24, 30, 20)M1,(17, 3, 13, 25, 31, 1)M1, (29, 9, 11, 30, 16, 27)M1,andA.D. FORBES ET AL. /AUSTRALAS. J. COMBIN. 70 (3) (2018), 386–389 389(16, 8, 9, 3, 18, 25)M2, (27, 5, 8, 6, 30, 0)M2, (0, 7, 20, 27, 1, 24)M2,(5, 3, 19, 20, 25, 1)M2, (22, 14, 20, 5, 0, 9)M2, (31, 20, 21, 18, 19, 24)M2,(14, 8, 11, 25, 1, 4)M2, (8, 17, 26, 19, 23, 0)M2, (8, 18, 30, 19, 12, 2)M2,(27, 11, 15, 2, 1, 22)M2, (0, 13, 14, 15, 4, 28)M2, (19, 10, 14, 2, 1, 8)M2,(16, 7, 14, 30, 17, 0)M2, (26, 9, 18, 14, 1, 15)M2, (31, 14, 23, 17, 2, 9)M2,(0, 21, 25, 14, 15, 29)M2, (19, 0, 11, 31, 27, 2)M2, (16, 10, 12, 0, 20, 1)M2,(21, 2, 26, 0, 7, 12)M2, (12, 5, 18, 0, 14, 6)M2, (6, 15, 19, 21, 9, 1)M2,(4, 13, 16, 19, 6, 25)M2, (15, 16, 22, 31, 17, 4)M2, (28, 16, 23, 11, 14, 26)M2,(16, 21, 24, 3, 27, 29)M2, (26, 5, 23, 15, 12, 20)M2, (3, 15, 28, 18, 14, 27)M2,(27, 12, 24, 15, 6, 23)M2, (5, 4, 21, 27, 10, 30)M2, (10, 27, 31, 25, 18, 4)M2,(2, 17, 18, 27, 20, 24)M2, (31, 4, 12, 9, 8, 29)M2, (3, 7, 31, 26, 18, 23)M2,(7, 6, 18, 31, 19, 4)M2, (24, 18, 22, 13, 4, 8)M2, (28, 13, 31, 5, 8, 21)M2,(13, 2, 6, 28, 20, 8)M2, (29, 1, 18, 11, 6, 20)M2, (10, 6, 24, 26, 22, 8)M2,(7, 22, 25, 26, 8, 15)M2, (30, 4, 26, 11, 6, 16)M2, (28, 19, 25, 12, 9, 27)M2,(4, 22, 29, 19, 20, 23)M2, (29, 2, 3, 4, 10, 26)M2, (29, 23, 25, 2, 13, 21)M2,(23, 21, 22, 6, 9, 19)M2, (28, 22, 30, 2, 11, 20)M2, (17, 12, 22, 3, 20, 10)M2,(29, 12, 28, 1, 14, 24)M2, (13, 1, 7, 2, 11, 12)M2, (13, 10, 23, 1, 24, 30)M2,(3, 1, 30, 21, 6, 0)M2, (3, 10, 11, 21, 16, 26)M2, (5, 9, 17, 21, 16, 2)M2,(17, 7, 29, 5, 31, 1)M2, (5, 24, 28, 17, 11, 22)M2, (9, 7, 10, 28, 15, 20)M2,(23, 7, 24, 11, 30, 12)M2, (30, 11, 17, 6, 31, 15)M2, (13, 9, 25, 24, 26, 27)M2,(25, 3, 17, 13, 10, 15)M2, (30, 9, 29, 11, 20, 25)M2.References[1] P. Adams, D.E. Bryant and M. Buchanan, A survey on the existence of G-designs, J. Combin. Des. 16 (2008), 373–410.[2] D.E. Bryant and T.A. McCourt, Existence results for G-designs,http://wiki.smp.uq.edu.au/G-designs/.[3] C. J. Colbourn and A. Rosa, Triple Systems, Clarendon Press, Oxford, 1999.[4] A.D. Forbes and T. S. Griggs, Designs for graphs with six vertices and nine edges,Australas. J. Combin. 70 (2018), 52–74.[5] Qing-de Kang, Lan-dang Yuan and Shu-xia Liu, Graph Designs for all Graphswith Six Vertices and Eight Edges, Acta Mathematicae Applicatae Sinica, EnglishSeries 21 (2005), 469–484.[6] D.R. Stinson, Hill-climbing algorithms for the construction of combinatorial de-signs, Ann. Discrete Math. 26 (1985), 321–334.(Received 20 July 2017)

Completing the design spectra for graphs with six vertices and eight edges

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