Group divisible designs GDD(n, n, n, 1; λ 1,λ 2 )

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 69(1) (2017), Pages Group divisible desigs GDD(,,, 1; λ 1,λ 2 ) Atthakor Sakda Chariya Uiyyasathia Departmet of Mathematics ad Computer Sciece Faculty of Sciece, Chulalogkor Uiversity Bagkok Thailad atthakor.sak@sru.ac.th chariya.u@chula.ac.th Abstract We give a complete solutio for the existece problem of group divisible desigs (or PBIBDs) with blocks of size k = 3, four groups of size (,,, 1), ad ay two idices (λ 1,λ 2 ). Moreover, we itroduce a costructio of ifiitely may group divisible desigs with t groups of size ad oe group of size 1. The costructio techique utilizes our mai result, together with some other kow desigs. 1 Itroductio A group divisible desig GDD(g = g 1 + g g s,s,k; λ 1,λ 2 ) is a ordered pair (G, B) whereg is a g-set of symbols that is partitioed ito s sets, called groups, of size g 1,g 2,...,g s,adb a collectio of k-subsets of G, called blocks, such that each pair of symbols from the same group appear together i λ 1 blocks ad each pair of symbols from distict groups appear together i λ 2 blocks. λ 1 ad λ 2 are called the idices of the desig. A group divisible desig is a partially balaced icomplete block desig (PBIBD) where the set of symbols are partitioed ito groups with two differet associates. Symbols occurrig together i the same group are called first associates, ad symbols occurrig i differet groups are called secod associates. (See [2, 3].) May papers i the literature have focused o the desigs with k = 3. Fu, Rodger, ad Sarvate [2, 3] completely solved the existece of group divisible desigs where all groups have the same size, amely GDD(g = + + +, m, 3; λ 1,λ 2 ). I 1992, Colbour, Hoffma, ad Rees [1] showed a ecessary ad sufficiet coditio for the existece of a GDD(g = u, t +1, 3; 0, 1). Later, Pabhapote ad Puim [9] ivestigated all triples of positive itegers (, m, λ) forwhicha GDD(g = + m, 2, 3; λ, 1) exists. Pabhapote [8] proved the existece of GDD(g = Correspodig author.

2 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), m, 2, 3; λ 1,λ 2 ) for all m 2ad 2 i which λ 1 λ 2. I 2014, Lapchida et al. [5, 6] worked o GDDs with three groups; they gave a complete solutio for the existece problem of group divisible desigs with block size k = 3 ad three groups of size (,, 1) i two separated cases λ 1 λ 2 ad λ 1 <λ 2.Herewegiveacomplete solutio for the group divisible desigs with block size k = 3 ad four groups of size (,,, 1) for ay two idices (λ 1,λ 2 ). Havig three groups of the same size allows us to utilize lati squares i our costructio techique. I the last sectio, we exted the mai result to costruct ifiitely may GDD(g = + ++1,t+1, 3; λ 1,λ 2 )s. Sice for the mai result we are dealig with GDDs with four groups ad block size 3, the otatio GDD(,,, 1; λ 1,λ 2 )isusedforgdd(g = , 4, 3; λ 1,λ 2 ) from this poit forward, ad we refer to blocks as triples. Our ecessary coditios for the existece problem of a GDD(,,, 1; λ 1,λ 2 ) ca be easily obtaied from a graph model by describig a GDD(,,, 1; λ 1,λ 2 ) graphically as follows. Let λk v deote the graph of v vertices where each pair of vertices is joied by λ edges. Let G 1 ad G 2 be graphs. The graph G 1 λ G 2 is obtaied from the uio of G 1 ad G 2 by joiig each vertex i G 1 to each vertex i G 2 with λ edges. A G-decompositio of a graph H is a partitio of the edges of H such that each elemet of the partitio iduces a copy of G. The existece of a GDD(,,, 1; λ 1,λ 2 ) is easily see to be equivalet to the existece of a K 3 -decompositio of λ 1 K λ2 (λ 1 K λ2 (λ 1 K λ2 K 1 )). As i [2], edges joiig vertices i the same group are called pure edges, otherwise,they are cross edges. Theorem 1.1. (Necessary Coditios) Let 1 ad λ 1,λ 2 0 be itegers. If there exists a GDD(,,, 1; λ 1,λ 2 ),the (i) 2 ( 1)λ 1 + λ 2,ad (ii) if =2the λ 1 3λ 2. Proof. Let G = λ 1 K λ2 (λ 1 K λ2 (λ 1 K λ2 K 1 )). Sice there exists a K 3 - decompositio of G, each vertex must have eve degree. Vertices of G are of degree 3λ 2 or ( 1)λ 1 +(2 +1)λ 2, so this yields (i). (Note that the vertex degree also yields 2 3λ 2. However, (i) has already implied 2 3λ 2.) Whe = 2, ay pure edge must be cotaied i a triple which cotais exactly two cross edges. Thus the umber of pure edges is at most half of the umber of cross edges, ad so (ii) holds. 2 Prelimiary Backgroud This sectio icludes the major tools that are used i our costructio of GDDs, amely lati squares, triple systems, ad packigs. Our lati squares of order are always based o the symbol set {1, 2, 3,...,}. If L = {l ij } is a lati square, we refer to l ij as the symbol i the cell (i, j) ofl. L = {l ij } is idempotet if l ii = i for all i. TwolatisquaresA = {a ij } ad B = {b ij } of the same order are called orthogoal if the 2 ordered pairs (a ij,b ij ), the pairs formed by superimposig oe square o the other, are all differet. The existece

3 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), of idempotet lati squares ad orthogoal lati squares are well-kow, as give i Theorem 2.1; see [10]. Theorem 2.1. [10] (i) For all positive itegers 2, there exists a idempotet lati square of order. (ii) For all positive itegers 2, 6, there exists a pair of orthogoal lati squares of order. A triple system TS(, λ) of idex λ ad order is a ordered pair (S, T ), where S is a -set, ad T is a collectio of 3-subsets of S called triples or blocks, such that each pair of distict elemets of S appear together i λ triples. We ca cosider a triple system TS(, λ) tobeagdd(g =, 1, 3; λ, λ 2 )oragdd(g = ,,3; λ 1,λ 2 )whereλ 1 ad λ 2 are ay oegative itegers. The existece of triple systems is cocluded i Theorem 2.2. See more details i [7]. Theorem 2.2. [7] A TS(, λ) exists if ad oly if λ ad satisfy oe of the followig cases: (i) λ 0(mod6)for all positive itegers 2, (ii) λ 1 or 5(mod6)for all positive itegers 1 or 3(mod6), (iii) λ 2 or 4(mod6))for all positive itegers 0 or 1(mod3),ad (iv) λ 3(mod6)for all odd positive itegers. A packig with triagles of the complete graph K is a 3-tuple (S, T, L), where S is the vertex set of K, T is a collectio of edge disjoit complete subgraphs K 3 of K,adL is the collectio of edges i K ot belogig to oe of the K 3 of T.The collectio of edges L is called the leave. If L is as small as possible, the (S, T, L) is called a maximum packig of order ; see [7]. Theorem 2.3. [7] Let be ay positive iteger. If (S, T, L) is a maximum packig of order, the the leave is (i) a 1-factor if 0 or 2(mod6), (ii) a 4-cycle if 5(mod6), (iii) a tripole, that is a spaig graph with each vertex havig odd degree ad cotaiig +2 edges, if 4(mod6),ad 2 (iv) the empty set if 1 or 3(mod6). 3 Sufficiecy Whe λ 2 =0,aGDD(,,, 1; λ 1, 0) exists if ad oly if there exists a TS(, λ 1 ). Hece we focus oly o GDDs with λ 2 1. Whe = 2, the extra ecessary coditio (iii) i Theorem 1.1 suggests that we costruct a GDD(2, 2, 2, 1; λ 1,λ 2 ) separately. Throughout the rest of the paper, for ay positive iteger we let X = {x 1,x 2,...,x }, Y = {y 1,y 2,...,y }, Z = {z 1,z 2,...,z } ad W = {w} be disjoit sets ad let V = X Y Z W.

4 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Lemma 3.1. Let, λ 2 1 ad λ 1 0 be itegers. There exists a GDD(2, 2, 2, 1; λ 1,λ 2 ) where 2 (λ 1 + λ 2 ) ad λ 1 λ 2. Proof. Let B 1 = {{w, x 1,y 2 }, {w, y 2,z 1 }, {w, z 1,x 2 }, {w, x 2,y 1 }, {w, y 1,z 2 }, {w, z 2,x 1 }, {x 1,y 1,z 1 }, {x 2,y 2,z 2 }, {x 1,y 1,z 1 }, {x 1,y 2,z 2 }, {x 2,y 1,z 2 }, {x 2,y 2,z 1 }}. The (V 2, B 1 )isagdd(2, 2, 2, 1; 0, 2). Sice 2 (λ 2 + λ 1 )adλ 1 λ 2,wehavethat λ 2 λ 1 is a eve oegative iteger. Let B be 1 2 (λ 2 λ 1 ) copies of B 1 ;the(v 2, B) is a GDD(2, 2, 2, 1; 0,λ 2 λ 1 ). By Theorem 2.2, we ca let (V 2, T )beats(7,λ 1 ), ad thus (V 2, B T) is our desired GDD. Lemma 3.2. Let, λ 2 1 ad λ 1 0 be itegers. There exists a GDD(2, 2, 2, 1; λ 1,λ 2 )where2 (λ 1 + λ 2 ) ad λ 2 <λ 1 3λ 2. Proof. Let B ={{w, x 1,x 2 }, {w, y 1,y 2 }, {w, z 1,z 2 }, {x 1,x 2,y 1 }, {x 1,x 2,y 2 }, {y 1,y 2,z 1 }, {y 1,y 2,z 2 }, {z 1,z 2,x 1 }, {z 1,z 2,x 2 }}. The (V 2, B) isagdd(2, 2, 2, 1; 3, 1). By Theorem 2.2, let (V 2, T )isats(7, 1). Sice 2 (λ 1 + λ 2 )adλ 2 <λ 1 3λ 2, write λ 1 λ 2 =2q where 0 q λ 2. The (V 2, C) wherec is the uio of q copies of B ad λ 2 q copies of T is a GDD(2,2,2,1;λ 1,λ 2 ). For 2, we first costruct GDD(,,, 1; λ 1,λ 2 )s where λ 1 λ 2, ad use them to costruct the GDDs for the remaiig case λ 1 >λ 2. Lemma 3.3 costructs the desigs cotaiig oly cross edges. Lemma 3.3. Let, λ 2 1 ad λ 1 0 be itegers. If 2ad λ 2 is eve, the there exists a GDD(,,, 1; 0,λ 2 ). Proof. It suffices to show oly whe λ 2 =2.Fori {1, 2,...,}, let B i = {{w, x i,y i }, {w, x i,z i }, {w, y i,z i }, {x i,y i,z i }}. By Theorem 2.1, there is a idempotet lati square L = {l ij } of order. Let B = {{x i,y j,z k } : i, j, k {1, 2,...,},l ij = k ad i j}. It is easily see that pairs of elemets from differet groups occur either twice i B i or precisely oce i B. The(V, B )isagdd(,,, 1; 0, 2), where B is the uio of B i, for all i {1, 2,...,} ad two copies of B. Lemmas 3.4, 3.5 ad 3.7 provide the costructio of a GDD(,,, 1; λ 1,λ 2 )where 2adλ 1 λ 2.

5 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Lemma 3.4. Let, λ 2 1 ad λ 1 0 be itegers. If 2is eve, λ 1 λ 2 ad λ 1 λ 2 (mod 2), the there exists a GDD(,,, 1; λ 1,λ 2 ). Proof. Sice is eve, the, by Theorem 2.2, there exists a TS(3 +1,λ 1 ), say (V, T ). Sice λ 1 λ 2 (mod 2), λ 2 λ 1 is eve. The, by Lemma 3.3, let (V, B) be a GDD(,,, 1; 0,λ 2 λ 1 ). Therefore (V, B T)isadesiredGDD. Lemma 3.5. Let, λ 2 1 ad λ 1 0 be itegers. If 1, 3(mod6),adλ 2 0 (mod 2), the there exists a GDD(,,, 1; λ 1,λ 2 ). Proof. We ca decompose pure edges ad cross edges separately. For pure edges, sice 1, 3 (mod 6), by Theorem 2.2, there exists a TS(, λ 1 ). Let (X, T 1 ), (Y, T 2 )ad(z, T 3 )bethosets(, λ 1 )s which exist from the theorem. For cross edges, sice λ 2 is eve, by Lemma 3.3, there exists a GDD(,,, 1; 0,λ 2 ), say (V, B). The (V, B T 1 T 2 T 3 )isadesiredgdd. Lemma 3.6. Let > 0 be a iteger. GDD(,,, 1; 1, 2). If 5(mod6), the there exists a Proof. Let (X, T 1, L 1 ), (Y, T 2, L 2 )ad(z, T 3, L 3 ) be maximum packigs with triagles of order. I additio, by Theorem 2.3 (ii), sice 5 (mod 6), the leaves L i are 4-cycles, say Let For i {5, 6,...,}, let L 1 = {{x 1,x 2 }, {x 2,x 3 }, {x 3,x 4 }, {x 4,x 1 }}, L 2 = {{y 1,y 2 }, {y 2,y 3 }, {y 3,y 4 }, {y 4,y 1 }}, L 3 = {{z 1,z 2 }, {z 2,z 3 }, {z 3,z 4 }, {z 4,z 1 }}. C 1 = {{w, x 1,x 2 }, {w, x 2,x 3 }, {w, x 3,x 4 }, {w, x 4,x 1 }}, C 2 = {{w, y 1,y 2 }, {w, y 2,y 3 }, {w, y 3,y 4 }, {w, y 4,y 1 }}, C 3 = {{w, z 1,z 2 }, {w, z 2,z 3 }, {w, z 3,z 4 }, {w, z 4,z 1 }}. B ii = {{x i,y i,z i }, {w, x i,y i }, {w, y i,z i }, {w, x i,z i }}. By Theorem 2.1, there is a idempotet lati square L = {l ij } of order. For i j {1, 2,...,} or i = j {1, 2, 3, 4}, let B ij = {{x i,y j,z lij }, {x i,y j,z lij }}. Let T be T 1 T 2 T 3, B the uio of B ij,fori, j {1, 2,...,}, adc the uio of C i,fori {1, 2, 3}. Thus(V, T B C)isaGDD(,,, 1; 1, 2). Lemma 3.7. Let, λ 2 1 ad λ 1 0 be itegers. If 5(mod6), λ 1 λ 2 ad λ 2 0(mod2), the there exists a GDD(,,, 1; λ 1,λ 2 ).

6 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Proof. If λ 1 0(mod3)theaTS(, λ 1 ) exists. Let (X, T 1 ), (Y, T 2 )ad(z, T 3 ) be TS(, λ 1 )s. Sice λ 2 is eve, by Lemma 3.3, let (V, B) beagdd(,,, 1; 0,λ 2 ). The (V, B T 1 T 2 T 3 ) is the desired GDD. If λ 1 1(mod3)theλ (mod 3). The there exists a GDD(,,, 1; λ 1 1,λ 2 2); together with a GDD(,,, 1; 1, 2) i Lemma 3.6, we have the desired GDD. Similarly, a GDD(,,, 1; λ 1 2,λ 2 2) exists where λ 1 2 (mod 3), ad by Theorem 2.2, there always exists a TS(3 +1, 2) which is a GDD(,,, 1; 2, 2); combiig them together yields our desired GDD. Theorem 3.8. Let, λ 2 1 ad λ 1 0 be itegers. If (, λ 1,λ 2 ) satisfy the ecessary coditios i Theorem 1.1 ad λ 1 λ 2, the there exists a GDD(,,, 1; λ 1,λ 2 ). Proof. If (, λ 1,λ 2 ) satisfies the ecessary coditios i Theorem 1.1, the λ 1 λ 2 (mod 2) where is eve, ad λ 2 0(mod2)where is odd. Therefore, by Lemmas 3.1, 3.4, 3.5 ad 3.7, there exists a GDD(,,, ; λ 1,λ 2 )whereλ 1 λ 2 Lemma 3.9. Let be ay positive iteger. If 2is eve, the there exists a GDD(,,, 1; 3, 1). Proof. If 0, 4 (mod 6) the, by Theorem 2.2, let (X, T 1 ), (Y, T 2 )ad(z, T 3 ) be TS(, 2)s ad (V, T )beats(3 +1, 1). The (V, T T 1 T 2 T 3 )isadesired GDD. If 2 (mod 6) the, by Theorem 2.3, let (X, T 1, L 1 ), (Y, T 2, L 2 )ad(z, T 3, L 3 ) be maximum packigs with triagles of order ; theleavesl t, t {1, 2, 3} are 1-factors. Let L 1 = {{x 1,x (1+ 2 ) }, {x 2,x (2+ 2 ) },...,{x ( 2 ),x }}, L 2 = {{y 1,y (1+ 2 ) }, {y 2,y (2+ 2 ) },...,{y ( 2 ),y }}, L 3 = {{z 1,z (1+ 2 ) }, {z 2,z (2+ 2 ) },...,{z ( 2 ),z }}. Sice 2 2, by Theorem 2.1, there is a idempotet lati square of order 2,sayL = {l ij }.Fori {1, 2,..., 2 },let ad for i j {1, 2,..., 2 },let B ii = {{w, x i,x (i+ 2 ) }, {w, y i,y (i+ 2 ) }, {w, z i,z (i+ 2 ) }, {x i,y i,z i }, {x i,y (i+ 2 ),z (i+ 2 ) }, {x (i+ 2 ),y i,z (i+ 2 ) }, {x (i+ 2 ),y (i+ 2 ),z i }}, B ij = {{x i,y j,z lij }, {x i,y (j+ 2 ),z (lij + 2 ) }, {x (i+ 2 ),y j,z (lij + 2 ) }, {x (i+ 2 ),y (j+ 2 ),z lij }}. Thus (V, T B), where T is the uio of three copies of T 1 T 2 T 3,adB is the uio of B ij for all i, j {1, 2,..., },isagdd(,,, 1; 3, 1). 2

7 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Lemma Let >0 be a iteger. If 2, the there exists a GDD (,,, 1; 5, 1). Proof. If 0, 4 (mod 6) the, by Theorem 2.2, let (X, T 1 ), (Y, T 2 )ad(z, T 3 ) be TS(, 4)s ad (V, T )beats(3 +1, 1). The (V, T T 1 T 2 T 3 )isadesired GDD. If 2 (mod 6), the let (X, T 1, L 1 ), (Y,T 2, L 2 )ad(z, T 3, L 3 )bemaximum packigs with triagles of order. The by Theorem 2.3, the leaves L t, t {1, 2, 3} are 1-factors. Let L 1 = {{x 1,x (1+ 2 ) }, {x 2,x (2+ 2 ) },...,{x ( 2 ),x }}, L 2 = {{y 1,y (1+ 2 ) }, {y 2,y (2+ 2 ) },...,{y ( 2 ),y }}, L 3 = {{z 1,z (1+ 2 ) }, {z 2,z (2+ 2 ) },...,{z ( 2 ),z }}. Sice is either 2 or 6, by Theorem 2.1 we ca let L = {l 2 ij} ad L = {lij } be two orthogoal lati squares of order. For each pair of i, j {1, 2,..., }, we defie 2 2 B ij i three cases separately. For i, j such that lij =1,letB ij be the collectio of blocks of a GDD(2,2,2,1;3,1) based o the symbol set {x i,x (i+ 2 ) } {y j,y (j+ 2 ) } {z lij,z (lij + 2 ) } {w}. For i, j such that lij =2, let B ij = {{x i,x (i+ 2 ),y j }, {x i,x (i+ 2 ),y (j+ 2 ) }, {y j,y (j+ 2 ),z lij }, {y j,y (j+ 2 ),z (lij + 2 ) }, {z lij,z (lij + 2 ),x i }, {z lij,z (lij + 2 ),x (i+ 2 ) }}. For i, j such that lij 3, let B ij = {{x i,y j,z lij }, {x i,y (j+ 2 ),z (lij + 2 ) }, {x (i+ 2 ),y j,z (lij + 2 ) }, {x (i+ 2 ),y (j+ 2 ),z lij }}. Let T be the uio of five copies of T 1 T 2 T 3 ad B the uio of B ij, for all i, j. Thus (V, T B)isaGDD(,,, 1; 5, 1). The ext mai theorem is the coclusio for the existece of our desired GDDs. Theorem Let >0 ad λ 1,λ 2 0 be itegers. There exists a GDD(,,, 1; λ 1,λ 2 )ifadolyif (i) λ 2 =0, ad there exists a TS(, λ 1 ), (ii) λ 2 0,=2, 2 (λ 1 + λ 2 ) ad λ 1 3λ 2,or (iii) λ 2 0, 2,ad2 ( 1)λ 1 + λ 2. Proof. For ecessity, (i) holds by Theorem 2.2, ad the other coditios hold by Theorem 1.1. Now we prove the sufficiecy. If we assume (i), the the statemet is true trivially. If we assume (ii), the the sufficiecy has bee proved by Lemmas Next we assume (iii); let 2adλ 2 0. Theorem 3.8 provides the case λ 1 λ 2. Assume that λ 1 > λ 2 ad write λ 1 = 6q + r where 0 r < 6. If

8 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), r λ 2, by Theorem 3.8 we ca let (V, B) beagdd(,,, 1; r, λ 2 ). Let (X, T 1 ), (Y, T 2 )ad(z, T 3 )bets(, 6q)s. The (V, B T 1 T 2 T 3 )isadesiredgdd. Now suppose that r > λ 2. The whe is eve, (r, λ 2 ) ca oly be oe of the ordered pairs i {(3, 1), (4, 2), (5, 1), (5, 3)}; adwhe is odd, (r, λ 2 ) ca oly be i {(3, 2), (4, 2), (5, 2), (5, 4)}. If is eve, a GDD(,,, 1; 3, 1) ad a GDD(,,, 1; 5, 1) are costructed i Lemmas 3.9 ad 3.10, respectively. Furthermore, a GDD(,,, 1;4, 2)ad a GDD(,,, 1; 5, 3) ca be costructed by addig oe ad two copies of a TS(3 +1, 1) to a GDD(,,, 1; 3, 1), respectively. If is odd, by Lemma 3.5 ad 3.6, let (V, B 1 )beagdd(,,, 1; 1, 2), ad by Lemma 3.3, let (V, B 2 )beagdd(,,, 1; 0, 2). By Theorem 2.2, let (X, T 1 ), (Y, T 2 )ad(z, T 3 )bets(, 3)s, ad (V, T 4 )beats(3 +1, 2). Therefore (V, B) is a GDD(,,, 1; r, λ 2 ), where (r, λ 2, B) is as i the followig table. (r, λ 2 ) B (3, 2) B 2 T 1 T 2 T 3 (4, 2) B 1 T 1 T 2 T 3 (5, 2) T 1 T 2 T 3 T 4 (5, 4) B 2 T 1 T 2 T 3 T 4 4 GDD(g = ,t+1, 3; λ 1,λ 2 ) I this sectio, we itroduce a costructio of ifiitely may GDD(g = ,t+1, 3; λ 1,λ 2 )s that utilizes our result ad the results from [5] ad [6], amely the existece of a GDD(g = , 4, 3; λ 1,λ 2 )adagdd(g = + +1, 3, 3; λ 1,λ 2 ). The ecessary coditios to obtai the desired GDDs are show i Theorem 4.1 which ca be proved by a stadard idea, similar to the proof of Theorem 1.1. Theorems 4.2 ad 4.3 form a complete solutio for the existece of a GDD(g = + +1, 3, 3; λ 1,λ 2 ). Beside these two theorems, our costructio also eeds Theorems 4.4 ad 4.5 to obtai a K 3 -decompositio of the graph λ 2 K t() ad λ 2 K t(),m where K t() is a complete multipartite graph with t groups of size ad K t(),m is a complete multipartite graph with t groups of size ad oe group of size m; see more details i [1] ad [4]. Theorem 4.1. (Necessary Coditios) Let, t, λ 1 ad λ 2 be itegers. If there exists a GDD(g = ,t+1, 3; λ 1,λ 2 ) if ad oly if (i) 2 λ 2 t, (ii) 2 [λ 1 ( 1) + λ 2 ((t 1) +1)], (iii) 3 t[λ 1 ( 1) + λ 2 ((t 1) + 2)], ad (iv) if =2the λ 1 tλ 2.

9 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Theorem 4.2. [5] Let, λ 1,λ 2 be positive itegers. If λ 1 <λ 2 the there exists a GDD(g = + +1, 3, 3; λ 1,λ 2 ) if ad oly if (i) 2 [λ 1 ( 1) + λ 2 ( +1)], (ii) 3 [λ 1 ( 1) + λ 2 ( +2)],ad (iii) λ 2 2λ 1. Theorem 4.3. [6] Let 3,λ 1,λ 2 be positive itegers. If λ 1 λ 2 the there exists a GDD(g = + +1, 3, 3; λ 1,λ 2 ) if ad oly if (i) 2 [λ 1 ( 1) + λ 2 ( +1)],ad (ii) 3 [λ 1 ( 1) + λ 2 ( +2)]. Theorem 4.4. [4] Let, t ad λ 2 be positive itegers. If 2the there exists a GDD(g = , t, 3; 0,λ 2 ) if ad oly if (i) 2 λ 2 (t 1) (ii) 3 λ 2 t(t 1) 2,ad (iii) t 3. Theorem 4.5. [1] Let, m, t 1 be itegers. There exists a GDD(g = m, t +1, 3; 0, 1) if ad oly if (i) If t =2the m =, (ii) m (t 1), (iii) 2 [(t 1) + m] (iv) 2 t, ad (v) 3 [t(t 1) 2 + tm]. Now we will costruct ifiitely may GDD(g = ,t+1, 3; λ 1,λ 2 )s usig Theorems , together with Theorem The costructio is separated ito two corollaries depedig o the parity of the umber of groups of size i the desig. For our costructio, let X i = {x i1,x i2,...,x i } for i =1, 2,...,t ad W = {w} be disjoit sets. Corollary 4.6. Let 3,λ 1 ad λ 2 be positive itegers, ad let t 6 be a eve iteger. If there exists a GDD(g = + +1, 3, 3; λ 1,λ 2 ),ad, t, λ 1,λ 2 satisfy the ecessary coditios i Theorem 4.1, the there exists a GDD(g = ,t+1, 3; λ 1,λ 2 ). Proof. By the assumptio, for i =1, 2,..., t, we ca costruct a GDD(g = , 3, 3; λ 1,λ 2 ) based o the elemet set X 2i 1 X 2i W. Let H be the graph that represets such costructio for all i. Suppose that a K 3 -decompositio of graph G represets a GDD(g = ,t+1, 3; λ 1,λ 2 ). The G = H + λ 2 K t 2 (2). By Theorems , the degree of each vertex of both graphs G ad H is eve ad the umber of edges of graphs G ad H are both divisible by three. These yield that the degree of each vertex of the graph K t 2 (2) is also eve, ad the umber of edges of the graph K t 2 (2) is also divisible by three. Moreover, t 3sicet 6. By 2 Theorem 4.4, the graph K t 2 (2) ca be decomposed ito triagles. Hece there exists a desired GDD.

10 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), Corollary 4.7. Let 3,λ 1 ad λ 2 be positive itegers, ad let t 9 be a odd iteger. If there exist a GDD(g = + +1, 3, 3; λ 1,λ 2 ) ad a GDD(g = , 4, 3; λ 1,λ 2 ),ad, t, λ 1,λ 2 satisfy the ecessary coditios i Theorem 4.1, the there exists a GDD(g = ,t+1, 3; λ 1,λ 2 ). Proof. We costruct a GDD(g = + +1, 3, 3; λ 1,λ 2 ) based o the elemet set X 2i 1 X 2i W for i =1, 2,..., t 3, represeted by a K 2 3-decompositio of the graph H 1,adaGDD(g = , 4, 3; λ 1,λ 2 ) based o the elemet set X t 2 X t 1 X t W, represeted by a K 3 -decompositio of the graph H 2. We costruct our desired GDD similarly to the previous Corollary that is a K 3 -decompositio of the graph G = H 1 + H 2 + λ 2 K t 3 2 (2),3 ad use of Theorem 4.5 to obtai a K 3- decompositio of the graph λ 2 K t 3 (2),3. 2 Ackowledgemets We would like to thak all the aoymous reviewers for their very helpful isights. I particular, their suggestio to add the last sectio is very much appreciated. Their commets greatly improved the mauscript. Refereces [1] C. J. Colbour, D. G. Hoffma ad R. Rees, A ew class of group divisible desigs with block size three, J. Combi. Theory Ser. A 59 (1992), [2] H. L. Fu ad C. A. Rodger, Group divisible desigs with two associate classes =2orm =2,J. Combi. Theory Ser. A 83 (1998), [3] H.L. Fu, C. A. Rodger ad D. G. Sarvate, The existece of group divisible desigs with first ad secod associates, havig block size 3, Ars Combi. 54 (2000), [4] H. Haai, Balaced icomplete block desigs ad related desigs, Discrete Math. 11 (1975), [5] W. Lapchida, N. Puim ad N. Pabhapote, GDDs with two associate classes with three groups of sizes 1,, ad λ 1 <λ 2, Lect. Notes Comp. Sci (2013), [6] W. Lapchida, N. Puim ad N. Pabhapote, GDDs with two associate classes with three groups of sizes 1, ad, Australas. J. Combi. 58(2) (2014), [7] C. C. Lider ad C. A. Rodger, Desig Theory, CRC Press, Boca Rato, FL (1997).

11 A. SAKDA ET AL. / AUSTRALAS. J. COMBIN. 69 (1) (2017), [8] N. Pabhapote, Group divisible desigs with two associate classes ad with two uequal groups, It. J. Pure Appl. Math. 81(1) (2012), [9] N. Pabhapote ad N. Puim, Group divisible desigs with two associate classes ad λ 2 =1,It. J. Math. Math. Sci. Art (2011), 10 pp. [10] W. D. Wallis, Itroductio to Combiatorial Desigs, Chapma & Hall / CRC Press, New York (2007) (Received 5 Sep 2015; revised 22 Ja 2016, 7 Jue 2017)