On the Spetr of Biprtite Direte Sugrphs of K 4 R. C. Bunge, 1 S. I. El-Znti, 1, H. J. Fry, 1 K. S. Kruss, 2 D. P. Roerts, 3 C. A. Sullivn, 4 A. A. Unsiker, 5 N. E. Witt 6 1 Illinois Stte University, Norml, IL 61790 2 Alm College, Alm, MI 48801 3 Illinois Wesleyn University, Bloomington, IL 61701 4 Bowling Green Stte University, Bowling Green, OH 43403 5 Brighm Young University, Provo, UT 84602 6 Simeon Creer Aemy, Chigo, IL 60620 Astrt The omplete irete grph of orer n, enote K n, is the irete grph on n verties tht ontins the rs (u, v) n (v, u) for every pir of istint verties u n v. For given irete grph, D, the set of ll n for whih K n mits D-eomposition is lle the spetrum of D. In this pper, we fin the spetrum for eh iprtite sugrph of K 4 with 5 or fewer rs. 1 Introution One of the primry questions in the fiel of grph eompositions is Given grph G, whih omplete grphs mit G-eomposition? Answering this question for ll grphs of smll orer hs een the topi of vrious ppers. In prtiulr, this question ws nswere for grphs n uniform hypergrphs of smll orer in [3] n [5], respetively. In this pper we follow this line of inquiry to irete grphs y lssifying the spetr of iprtite irete grphs of orer t most 4, with up to 5 rs. If n re integers, we enote {, +1,..., } y [, ] (if >, then [, ] = ). Let N enote the set of nonnegtive integers n Z m the group of integers moulo m. Throughout this pper, we refer to irete grphs Reserh supporte y Ntionl Siene Fountion Grnt No. A1359300
s igrphs. For igrph, D, let V (D) n E(D) enote the vertex set of D n the r set of D, respetively. The orer n the size of igrph D re V (D) n E(D), respetively. Throughout this pper we will use the nmes for igrphs, isplye in Tles 1 n 2, foun in An Atls of Grphs [13] y Re n Wilson. Let H n D e igrphs suh tht D is sugrph of H. A D- eomposition of H is set = {D 1, D 2,..., D r } of pirwise ege-isjoint sugrphs of H eh of whih is isomorphi to D n suh tht E(H) = r i=1 E(D i). If D-eomposition of H exists, then we sy tht D eomposes H. The reverse orienttion of D is the igrph with vertex set V (D) n r set {(v, u): (u, v) E(D)}. Let D n H enote the reverse orienttions of D n H, respetively. We note tht the existene of D- eomposition of H neessrily implies the existene of D -eomposition of H. A D-eomposition of Kn is lso known s (Kn, D)-esign. The set of ll n for whih Kn mits D-eomposition is lle the spetrum of D. Sine Kn is its own reverse orienttion, we note tht the spetrum of D is equivlent to the spetrum of D. The neessry onitions for igrph D to eompose Kn inlue (A) V (D) n, (B) E(D) ivies n(n 1), n (C) g{outegree(v): v V (D)} n g{inegree(v): v V (D)} oth ivie n 1. There re 51 iprtite sugrphs of K 4 with no isolte verties. It ws shown in [6] tht 37 of these eompose K mx+1 ylilly (efine in Setion 2), where m is the numer of rs in the sugrph. Until now, the spetr for the mjority of these grphs h not een lssifie. This pper extens those results y fining the spetr for ll 42 iprtite sugrphs of K 4 with 5 rs or fewer. If D is sugrph of K 4 with m 5 rs, then y onition (B) the existene of (K n, D)-esign neessittes tht n 0 or 1 (mo m). Through series of lemms, we prove the following. Theorem 1. The spetr for the 42 iprtite sugrphs of K4 fewer rs re s isplye in Tle 3. with 5 or The spetr for ertin sugrphs (oth iprtite n non-iprtite) of K 4 hve lrey een stuie. When D is yli orienttion of K 3, then (K n, D)-esign is known s Menelsohn triple system. The spetrum for Menelsohn triple systems ws foun inepenently y Menelsohn [11] n Bermon [2].When D is trnsitive orienttion of K 3, then (K n, D)- esign is known s trnsitive triple system. The spetrum for trnsitive triple systems ws foun y Hung n Menelsohn [9]. In [8], Hrtmn n Menelsohn foun the spetr for ll remining simple onnete su- 2
grphs of K 3, whih inlue grphs D7, D10, D11, D13, D16, n D25 (see Tle 1). There re extly four orienttions of 4-yle. It ws shown in [15] tht if D is yli orienttion of 4-yle (i.e., D67), then (K n, D)- esign exists if n only if n 0 or 1 (mo 4) n n 4. The spetr for the remining three orienttions of 4-yle (i.e., D46, D59, D60) were foun in [7]. A irete pth on n verties, enote DP n, is the irete grph whose unerlying simple grph is pth on n verties suh tht every vertex tht is not lef in the unerlying grph hs oth inegree n outegree of one. In [12], neessry n suffiient onitions for eomposition of the omplete multi-igrph into irete pths of ritrrily presrie lengths were foun. In prtiulr, this hrterizes the spetr of D25 n D38. Here, we stte their result only for ege multipliity one n fixe pth length. Theorem 2. [12] Neessry n suffiient onitions for the existene of (K n, DP m+1 )-esign re m n 1 n n(n 1) 0 (mo m), unless (m, n) is either (4, 5) or (2, 3). A igrph is lle n ntiirete pth if its unerlying grph is pth, n it oes not mit irete pth of length 2 s sugrph. Let D e n ntiirete pth. Neessry n suffiient onitions for the existene of (K n, D)-esign were otine in [16]. Notie tht D7, D10, n D33 re ntiirete pths. Theorem 3. [16] Let AP m enote n ntiirete pth on m verties. A (Kn, AP m+1 )-esign exists if n only if the following onitions re stisfie: (1) m n 1, (2) n(n 1) 0 (mo m), (3) n or m is o. Let G e iprtite sugrph of the 2-fol unirete omplete grph on 4 verties. In [1] neessry n suffiient onitions were otine for the existene of G-eomposition of the 2-fol unirete omplete grph on n verties. Severl of these eompositions iretly trnslte to igrph eompositions of interest in this pper. In prtiulr, the spetr of D3, D4, D16, D27, D37, n D66 re otine from these results. 3
2 Grph lelings Let V (K n) = [0, n 1]. The length of n r (i, j) is j i if j > i, n it is n + j i otherwise. Note tht E(K n) onsists of n rs of length i for eh i [1, n 1]. Let D e sugrph of K n. By rotting D, we men pplying the permuttion i i + 1 to V (D) where the ition is one moulo n. Moreover in this se, if j N, then D + j is the igrph otine from D y suessively rotting D totl of j times. Note tht rotting n r oes not hnge its length. Also note tht D + j is isomorphi to D for every j N. A (K n, D)-esign is yli if rotting is n utomorphism of. Grph lelings were introue y Ros in [14] n provie mens of otining yli esigns for unirete grphs. In prtiulr, the wellknown greful leling ws efine in [14]. In 1985, Bloom n Hsu [4] extene the onept of greful leling to irete grphs. With the nottion pte to etter suit this pper, we present the following efinition from [4]. Let D e irete grph with m rs n t most m + 1 verties. Let f : V (D) [0, m] e n injetive funtion, n efine funtion f : E(D) [1, m] s follows: f((, )) = f() f(), if f() > f(), n f((, )) = m + 1 + f() f(), otherwise. We ll f irete ρ-leling of D if { f((, )): (, ) E(D)} = [1, m]. Thus, irete ρ-leling of D is n emeing of D in K m+1 suh tht there is extly one r in D of length i for eh i [1, m]. It ws shown in [10] tht for igrph D with size m n orer t most m + 1, yli (K m+1, D)-esign exists if n only if D mits irete ρ-leling. In orer to otin n infinite fmily of yli esigns, it is neessry to exten the notion of irete ρ-leling. In our se this is omplishe y restriting ourselves to iprtite igrphs n imposing n orer on the leling. Let D e iprtite irete grph with m rs n t most m + 1 verties. Let {A, B} e vertex iprtition of V (D). A irete ρ-leling f of D is orere if f() < f() for eh r with en verties A n B. An orere irete ρ-leling is lso lle irete ρ + -leling. The onnetion etween irete ρ + -lelings n yli igrph esigns is foun in result from [6]. Theorem 4. If D is iprtite irete grph with m rs tht mits irete ρ + -leling, then there exists yli (K mx+1, D)-esign for ll x Z +. Next we onsier n nlogue of 1-rottionl esigns for irete grphs. If we let V (K n) = [0, n 2] { }, then the length of n r (i, j), where {i, j}, is j i if j > i, n it is n 1 + j i otherwise. Furthermore, 4
we sy tht the length of n r of the form (i, ) is +, n the length of n r of the form (, j) is. Let D e sugrph of Kn. A (Kn, D)- esign is lle 1-rottionl irete esign if pplying the permuttion (0, 1, 2,..., n 2)( ) is n utomorphism of. Let D e irete grph with m rs n t most m verties. Let f : V (D) [0, m 1] { } e n injetive funtion n efine funtion f : E(D) [1, m 2] {+, } s follows: f() f() if f() > f(), m 1 + f() f() if f() < f(), f((, )) = + if f() =, if f() =. We ll f 1-rottionl irete ρ-leling of D if { f((, )): (, ) E(D)} = [1, m 2] {+, }. Thus, 1-rottionl irete ρ-leling of D is n emeing of D in K m suh tht there is extly one r in D of length i for eh i [1, m 2] {+, }. The following theorem formlizes this oservtion. Theorem 5. Let D e iprtite irete grph with m rs. There exists 1-rottionl (K m, D)-esign if n only if D mits 1-rottionl irete ρ-leling. 3 Min Results The 42 non-isomorphi iprtite sugrphs of K4 of size t most 5 re shown in Tles 1 n 2. These tles lso give key tht enotes lele opy for eh iprtite irete grph. For exmple, D11[,, ] refers to the grph with three verties lele,, n with two rs etween n n single r irete from to. Our generl metho for lssifying the spetrum for suh grph of size m is to rek into two ses: omplete grphs with mx + 1 verties n omplete grphs with mx verties. However, we first note the following negtive results for the suffiieny of the neessry onitions, whih n e esily verifie. Lemm 6. There oes not exist n (H, D)-esign for (H, D) {(K 4, D50), (K 5, D50), (K 4, D51), (K 6, D99)}. Let G n H e vertex-isjoint irete grphs. The join of G n H, enote G H, is efine to e the irete grph with vertex set V (G) V (H) n r set E(G) E(H) {(u, v), (v, u): u V (G), v V (H)}. We use the shorthn nottion t i=1 G i to enote G 1 G 2 G t, n when G i = Gj t = G for ll 1 i < j t, we use the nottion i=1 G. For exmple, K12 = 4 i=1 K 3. 5
Tle 1: Biprtite sugrphs of K 4 with 3 or fewer rs. 2 3 verties D3[, ] D4[, ] D7[,, ] D10[,, ] D11[,, ] D13[,, ] D25[,, ] D27[,,, ] D28[,,, ] D31[,,, ] D32[,,, ] 4 verties D33[,,, ] D36[,,, ] D37[,,, ] D38[,,, ] D39[,,, ] D40[,,, ] 6
Tle 2: Biprtite sugrphs of K 4 with 4 or 5 rs. D16[,, ] D41[,,, ] D44[,,, ] D46[,,, ] D50[,,, ] D51[,,, ] D52[,,, ] D54[,,, ] D56[,,, ] D59[,,, ] D60[,,, ] D62[,,, ] D63[,,, ] D64[,,, ] D66[,,, ] D67[,,, ] D70[,,, ] D77[,,, ] D84[,,, ] D86[,,, ] D91[,,, ] D92[,,, ] D99[,,, ] D100[,,, ] D104[,,, ] 7
Tle 3: The neessry n suffiient onitions for the given igrphs to eompose Kn. Dirph Conitions Referenes D3 n 1 [1] D4 n 1 [1] D7 n 1 (mo 2) [8], [16], [6] D10 n 1 (mo 2) [8], [16], [6] D11 n 0 or 1 (mo 3) [8], [6] D13 n 0 or 1 (mo 3) [8], [6] D16 n 0 or 1 (mo 4) [8], [6] D25 n = 1 or n 4 [8], [12] D27 n = 1 or n 4 [1] D28 n 1 (mo 3) [6] D31 n 0 or 1 (mo 3), n 3 [6], Lemm 12 D32 n 0 or 1 (mo 3), n 3 Lemms 8 n 13 D33 n 0 or 1 (mo 3), n 3 [16], [6] D36 n 0 or 1 (mo 3), n 3 [6], Lemm 12 D37 n 0 or 1 (mo 3), n = 1 or n 6 [1] D38 n 0 or 1 (mo 3), n 3 [12], [6] D39 n 0 or 1 (mo 3), n 3 Lemms 8 n 13 D40 n 1 (mo 3) [6] D41 n 0 or 1 (mo 4) [6], Lemm 14 D44 n 1 (mo 4) [6] D46 n 1 (mo 4) [7], [6] D50 n 0 or 1 (mo 4), n = 1 or n 8 Lemms 6, 9, n 15 D51 n 0 or 1 (mo 4), n 4 [6], Lemms 6 n 16 D52 n 0 or 1 (mo 4) Lemms 10 n 17 D54 n 0 or 1 (mo 4) [6], Lemm 18 D56 n 0 or 1 (mo 4) [6], Lemm 18 D59 n 0 or 1 (mo 4), n = 1 or n 8 [7] D60 n 0 or 1 (mo 4), n 5 [7] D62 n 0 or 1 (mo 4) [6], Lemm 14 D63 n 1 (mo 4) [6] D64 n 0 or 1 (mo 4) Lemms 10 n 17 D66 n 0 or 1 (mo 4) [6], [1] D67 n 0 or 1 (mo 4), n 4 [15] D70 n 0 or 1 (mo 5) [6], Lemm 19 D77 n 0 or 1 (mo 5) [6], Lemm 20 D84 n 0 or 1 (mo 5) [6], Lemm 21 D86 n 0 or 1 (mo 5) [6], Lemm 22 D91 n 0 or 1 (mo 5) [6], Lemm 19 D92 n 0 or 1 (mo 5) [6], Lemm 20 D99 n 0 or 1 (mo 5), n 6 Lemms 6, 11, n 23 D100 n 0 or 1 (mo 5) [6], Lemm 24 D104 n 0 or 1 (mo 5) [6], Lemm 21 8
3.1 Deompositions of K mx+1 Leling methos n e use to otin yli (K mx+1, D)-esigns for 31 of the 42 irete grphs. This is ue to the ft tht these grphs mit irete ρ + -lelings s shown in [6]. Theorem 7. [6] Let D e iprtite sugrph of K4 of size m, n let x e positive integer. If D oes not elong to the set {D25, D27, D32, D37, D39, D50, D52, D59, D60, D64, D99}, then there exists yli (Kmx+1, D)-esign. Next, we proee to the igrphs tht o not mit irete ρ + - leling. Throughout this setion we let the vertex set of Kmx+1 e [0, mx]. Furthermore, throughout the entirety of the pper let Ks,t hve vertex iprtition {A, B} where A = [0, s 1] n B = [s, s + t 1]. Lemm 8. For every integer x 1 there exists (K 3x+1, D)-esign for D {D32, D39}. Proof. Sine D39 is the reverse orienttion of D32, it suffies to show the existene of (K3x+1, D32)-esign. For x = 1, we hve the following (K4, D32)-esign: {D32[0, 3, 2, 1], D32[1, 2, 3, 0], D32[3, 2, 1, 0], D32[2, 3, 0, 1]}. For x > 1, we require the following (K 3,3, D32)-esign: {D32[0, 3, 4, 2], D32[1, 4, 5, 0], D32[2, 5, 3, 1], D32[4, 1, 0, 5], D32[5, 2, 1, 3], D32[3, 0, 2, 4]}. We write K3x+1 = K1 x i=1 K 3. On eh of the x opies of K1 K3, we ple (K4, D32)-esign. The remining rs form r-isjoint opies of K3,3, on eh of whih we ple (K3,3, D32)-esign. Lemm 9. For every integer x 2 there exists (K 4x+1, D50)-esign. Proof. For x = 2, onsier the igrph G = D50[0, 3, 5, 1] D50[0, 6, 4, 2]. It is esy to hek tht we hve irete ρ-leling of G, n thus (K 9, D50)-esign exists. For x = 3, onsier the irete grph H = D50[0, 6, 4, 1] D50[0, 4, 5, 2] D50[0, 5, 6, 3]. It is esy to hek tht we hve irete ρ-leling of H, n thus (K 13, D50)-esign exists. For x > 3, we require the following (K 4,4, D50)-esign: {D50[0, 5, 6, 4], D50[1, 6, 7, 5], D50[2, 7, 4, 6], D50[3, 4, 5, 7], D50[0, 6, 5, 7], D50[1, 7, 6, 4], D50[2, 4, 7, 5], D50[3, 5, 4, 6]}. 9
Cse 1: x = 2k for some integer k 2. We write K8k+1 = K1 k i=1 K 8. On eh of the k opies of K1 K8, we ple (K9, D50)-esign. The remining rs form r-isjoint opies of K8,8, eh of whih n e eompose into opies of K4,4. On eh of these opies of K4,4, we ple (K4,4, D50)-esign. Cse 2: x = 2k + 1 for some integer k 2. We write K 8k+5 = K 1 K 12 k 1 i=1 K 8. On the opy of K 1 K 12, we ple (K 13, D50)-esign. On eh of the k 1 opies of K 1 K 8, we ple (K 9, D50)-esign. The remining rs form r-isjoint opies of K 8,12 n K8,8, oth of whih n e eompose into opies of K4,4. On eh of these opies of K4,4, we ple (K4,4, D50)-esign. Lemm 10. For every integer x 1 there exists (K 4x+1, D)-esign for D {D52, D64}. Proof. Sine D64 is the reverse orienttion of D52, it suffies to show the existene of (K4x+1, D52)-esign. For x = 1, we hve the following (K5, D52)-esign: {D52[1, 0, 2, 3], D52[3, 4, 2, 1], D52[3, 1, 0, 4], D52[4, 1, 0, 2], D52[2, 4, 0, 3]}. For x > 1, we require the following (K 2,2, D52)-esign: {D52[0, 3, 2, 1], D52[1, 3, 2, 0]}. We write K4x+1 = K1 x i=1 K 4. On eh of the x opies of K1 K4, we ple (K5, D52)-esign. The remining rs form r-isjoint opies of K4,4, eh of whih n e eompose into opies of K2,2. On eh of these opies of K2,2, we ple (K2,2, D52)-esign. Lemm 11. For every integer x 2 there exists (K 5x+1, D99)-esign. Proof. For x = 2, we hve the following (K 11, D99)-esign: {D99[0, 1, 4, 7], D99[7, 3, 2, 6], D99[3, 0, 8, 5], D99[9, 5, 2, 4], D99[9, 8, 3, 6], D99[5, 10, 8, 3], D99[7, 8, 4, 1], D99[0, 2, 5, 1], D99[0, 6, 7, 5], D99[1, 3, 5, 2], D99[2, 10, 1, 6], D99[3, 4, 2, 5], D99[4, 6, 2, 8], D99[4, 9, 8, 1], D99[6, 7, 2, 9], D99[6, 9, 5, 3], D99[8, 0, 2, 7], D99[8, 10, 4, 0], D99[9, 0, 1, 10], D99[10, 4, 6, 1], D99[10, 7, 3, 5], D99[10, 9, 1, 7]}. For x = 3, onsier the igrph G = D99[0, 7, 2, 1] D99[0, 13, 8, 3] D99[0, 4, 5, 6]. It is esy to hek tht we hve irete ρ-leling of 10
G, n thus (K 16, D99)-esign exists. For x > 3, we require the following (K 5,10, D99)-esign: {D99[0, 5, 6, 2], D99[1, 6, 7, 3], D99[2, 7, 8, 4], D99[3, 8, 9, 0], D99[4, 9, 5, 1], D99[5, 3, 4, 14], D99[6, 4, 0, 10], D99[7, 0, 1, 11], D99[8, 1, 2, 12], D99[9, 2, 3, 13], D99[12, 0, 4, 8], D99[13, 1, 0, 9], D99[14, 2, 1, 5], D99[10, 3, 2, 6], D99[11, 4, 3, 7], D99[0, 14, 11, 1], D99[1, 10, 12, 2], D99[2, 11, 13, 3], D99[3, 12, 14, 4], D99[4, 13, 10, 0]}; n the rgument proees similrly s in the proof for Lemm 9. 3.2 Deompositions of K mx Neessry onition (A) implies tht there is no (K 3, D)-esign for ny igrph D {D28, D31, D32, D33, D36, D37, D38, D39, D40}. Furthermore, there is no (K 2, D27)-esign. Now, let D {D7, D10, D28, D40, D44, D63} n suppose D hs m rs. Then neessry onition (C) implies tht there is no (K mx, D)-esign for ny positive integer x. Throughout this setion we let V (K m) = [0, m 2] { }. Lemm 12. For every integer x 2 there exists (K 3x, D)-esign for D {D31, D36}. Proof. Sine D36 is the reverse orienttion of D31, it suffies to show the existene of (K 3x, D31)-esign. Let x 2 e n integer. The following is (K 3x, D31)-esign: {D31[0, 1,, 2], D31[0,, 1, 3]} {D31[0, 3 + 3(i 3), 4 + 3(i 3), 6 + 3(i 3)]: 3 i x}. Lemm 13. For every integer x 2 there exists (K 3x, D)-esign for D {D32, D39}. Proof. Sine D39 is the reverse orienttion of D32, it suffies to show the existene of (K 3x, D32)-esign. Let x 2 e n integer. The following is (K 3x, D32)-esign: {D32[0, 1,, 2]} {D32[0, 3 + 3(i 2), 4 + 3(i 2), 1]: 2 i x}. Lemm 14. For every integer x 1 there exists (K 4x, D)-esign for D {D41, D62}. 11
Proof. Sine D62 is the reverse orienttion of D41, it suffies to show the existene of (K 4x, D41)-esign. Let x 1 e n integer. The following is (K 4x, D41)-esign: {D41[0, 1, 2, ]} {D41[0, 1+4(i 1), 2+4(i 1), 4(i 1)]: 2 i x}. Lemm 15. For every integer x 2 there exists (K 4x, D50)-esign. Proof. Let x 2 e n integer. Cse 1: x = 2k for some integer k 1. The following is (K 8k, D50)- esign: {D50[0, 1, 2, ], D50[0, 6, 5, 4]} {D50[0, 1 + 8(i 1), 2 + 8(i 1), 8(i 1)]: 2 i x} {D50[0, 6 + 8(i 1), 5 + 8(i 1), 4 + 8(i 1)]: 2 i x}. Cse 2: x = 2k + 1 for some positive integer k 1. For k = 1, onsier the igrph G = D50[0, 7, 3, ] D50[0, 3, 5, 1] D50[0, 5, 7, 2]. It is esy to hek tht we hve 1-rottionl irete ρ-leling of G, n thus (K 12, D50)-esign exists. For k > 1, we write K 8k+4 = K 12 k 1 i=1 K 8. On the opy of K 12, we ple (K 12, D50)-esign. On eh of the k 1 opies of K 8, we ple (K 8, D50)-esign, whih is shown to exist in the proof of Cse 1. The remining rs form r-isjoint opies of K 8,12 n K 8,8, oth of whih n e eompose into opies of K 4,4. On eh of these opies of K4,4, we ple (K4,4, D50)-esign, whih is shown to exist in the proof for Lemm 9. Lemm 16. For every integer x 2 there exists (K 4x, D51)-esign. Proof. Let x 2 e n integer. The following is (K 4x, D51)-esign: {D51[0,, 1, 4], D51[0, 3, 2, ]} {D51[0, 6 + 4(i 3), 7 + 4(i 3), 2]: 3 i x}. Lemm 17. For every integer x 1 there exists (K 4x, D)-esign for D {D52, D64}. Proof. Sine D64 is the reverse orienttion of D52, it suffies to show the existene of (K4x, D52)-esign. For x = 1, onsier the igrph D52[0, 1,, 2]. It is esy to hek tht we hve 1-rottionl irete ρ- leling of D52, n thus (K4, D52)-esign exists. For x > 1, we write K 4x = x i=1 K 4. On eh of the x opies of K 4, we ple (K 4, D52)- esign. The remining rs form r-isjoint opies of K 4,4, eh of whih n e eompose into opies of K 2,2. On eh of these opies of K 2,2, we ple (K2,2, D52)-esign, whih is shown to exist in the proof for Lemm 10. 12
Lemm 18. For every integer x 1 there exists (K 4x, D)-esign for D {D54, D56}. Proof. Sine D56 is the reverse orienttion of D54, it suffies to show the existene of (K 4x, D54)-esign. Let x 1 e n integer. The following is (K 4x, D54)-esign: {D54[0,, 2, 1]} {D54[0, 4 + 4(i 2), 6 + 4(i 2), 1]: 2 i x}. Lemm 19. For every integer x 1 there exists (K 5x, D)-esign for D {D70, D91}. Proof. Sine D91 is the reverse orienttion of D70, it suffies to show the existene of (K 5x, D70)-esign. Let x 1 e n integer. The following is (K 5x, D70)-esign: {D70[0,, 2, 1]} {D70[0, 5(i 1), 2+5(i 1), 1+5(i 1)]: 2 i x}. Lemm 20. For every integer x 1 there exists (K 5x, D)-esign for D {D77, D92}. Proof. Sine D92 is the reverse orienttion of D77, it suffies to show the existene of (K 5x, D77)-esign. Let x 1 e n integer. The following is (K 5x, D77)-esign: {D77[0,, 3, 1]} {D77[0, 5(i 1), 3 + 5(i 1), 1]: 2 i x}. Lemm 21. For every integer x 1 there exists (K 5x, D)-esign for D {D84, D104}. Proof. Sine D104 is the reverse orienttion of D84, it suffies to show the existene of (K 5x, D84)-esign. Let x 1 e n integer. The following is (K 5x, D84)-esign: {D84[0,, 2, 1]} {D84[4 + 5(i 2), 0, 2, 5 + 5(i 2)]: 2 i x}. Lemm 22. For every integer x 1 there exists (K5x, D86)-esign. Proof. For x = 1, we hve the following (K5, D86)-esign: {D86[0, 1, 2, 3], D86[0, 3,, 2], D86[, 1, 0, 2], D86[, 3, 2, 1]}. For x > 1, we require the following (K5,5, D86)-esign: {D86[0, 5, 7, 4], D86[1, 6, 8, 0], D86[2, 7, 9, 1], D86[3, 8, 5, 2], D86[4, 9, 6, 3], D86[6, 2, 4, 5], D86[7, 3, 0, 6], D86[8, 4, 1, 7], D86[9, 0, 2, 8], D86[5, 1, 3, 9]}; n the rgument proees similrly s in the proof for Lemm 17. 13
Lemm 23. For every integer x 1 there exists (K 5x, D99)-esign. Proof. For x = 1, onsier the igrph D99[0, 2,, 1]. It is esy to hek tht we hve the 1-rottionl irete ρ-leling of D99, n thus (K 5, D99)-esign exists. For x = 2, We hve the following (K 10, D99)-esign: {D99[1, 0, 4, 5], D99[2, 0, 5, 6], D99[3, 0, 6, 4], D99[0, 7, 4, 1], D99[0, 8, 5, 2], D99[0, 9, 6, 3], D99[7, 4, 1, 5], D99[8, 5, 2, 6], D99[9, 6, 3, 4], D99[4, 8, 2, 7], D99[5, 9, 3, 8], D99[6, 7, 1, 9], D99[8, 1, 7, 5], D99[9, 2, 8, 6], D99[7, 3, 9, 4], D99[2, 1, 4, 9], D99[3, 2, 5, 7], D99[1, 3, 6, 8]}. For x > 2, we require (K 5,10, D99)-esign, whih is shown to exist in the proof for Lemm 11. Cse 1: x = 2k + 1 for some integer k 2. We write K 10k+5 = K 5 k i=1 K 10. On the opy of K 5, we ple (K 5, D9)- esign. On eh of the k opies of K 10, we ple (K 10, D99)-esign. The remining rs form r-isjoint opies of K 5,10 n K 10,10, whih n e eompose into opies of K 5,10. On eh of these opies of K 5,10, we ple (K 5,10, D99)-esign. Cse 2: x = 2k for some integer k 2. We write K 10k = k i=1 K 10. On eh of the k opies of K 10, we ple (K 10, D99)-esign. The remining rs form r-isjoint opies of K 10,10, eh of whih n e eompose into opies of K5,10. On eh of these opies of K5,10, we ple (K5,10, D99)-esign. Lemm 24. For every integer x 1 there exists (K 5x, D100)-esign. Proof. Let x 1 e n integer. The following is (K 5x, D100)-esign: {D100[0, 1, 2, ]} {D100[0, 1 + 5(i 1), 2 + 5(i 1), 2]: 2 i x}. Thus, we hve shown tht Theorem 1 hols. 4 Aknowlegments The uthors re grteful to Peter Ams for help in fining some of the smll esigns neee for this work. This reserh is supporte y grnt numer A1359300 from the Division of Mthemtil Sienes t the Ntionl Siene Fountion. This work ws one while ll ut the first, seon, n fifth uthors were prtiipnts in REU Site: Mthemtis Reserh Experiene for Pre-servie n for In-servie Tehers t Illinois Stte University. 14
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