On the Spectra of Bipartite Directed Subgraphs of K 4

Transcription

1 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 Alm College, Alm, MI Illinois Wesleyn University, Bloomington, IL Bowling Green Stte University, Bowling Green, OH Brighm Young University, Provo, UT Simeon Creer Aemy, Chigo, IL 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. A

2 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

3 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

4 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 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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 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

12 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

13 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

14 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 A 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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