Hamiltonian Connectedness of Toeplitz Graphs
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1 Hmiltonin Connectedne of Toeplitz Grph Muhmmd Fil Ndeem, Aeh Shir, nd Tudor Zmfirecu Introduction A imple undirected grph T with vertice ;;:::;n i clled Toeplitz grph if it djcenc mtri A.T / i Toeplitz. A Toeplitz mtri i n.n n/ mmetric mtri which h contnt vlue long ll digonl prllel to the min digonl. Therefore, Toeplitz grph T i uniquel defined the firt row of A.T /, (0 ) equence. If the in tht equence re plced t poition C t ;C t ;:::;C t k with t <t < <t k <n, we m impl write T D T n ht ;t ;:::;t k i,two vertice, eing connected n edge iff j j ft ;t ;:::;t k g. Let G e grph of order n. IticlledHmiltonin if it contin ccle of order n. It i clled trcele, if it contin pth of order n; tht pth i then clled Hmiltonin pth of G. The grph G i id to e Hmiltonin connected if for n pir of ditinct vertice u nd v of G, there eit Hmiltonin pth from u to v. The propert of eing Hmiltonin connected i tronger thn eing Hmiltonin. The third uthor work w upported grnt of the Romnin Ntionl Authorit for Scientific Reerch, CNCS UEFISCDI, project numer PN-II-ID-PCE M.F. Ndeem A. Shir Adu Slm School of Mthemticl Science, GC Univerit, 68-B, New Mulim Town, Lhore, Pkitn e-mil: mfilndeem@mil.com; hinori@hotmil.com T. Zmfirecu ( ) Fcult of Mthemtic, Univerit of Dortmund, 44 Dortmund, Germn Intitute of Mthemtic Simion Stoïlow Roumnin Acdem, Buchret, Roumni Adu Slm School of Mthemticl Science, GC Univerit, 68-B, New Mulim Town, Lhore, Pkitn e-mil: tuzmfirecu@googlemil.com. Springer Bel 05 P. Crtier et l. (ed.), Mthemtic in the t Centur, Springer Proceeding in Mthemtic & Sttitic 98, DOI 0.007/
2 36 M.F. Ndeem et l. Reference [ 4] contin reult out connectivit, iprtitene, plnrit, nd colorilit of Toeplitz grph. Some Hmiltonin propertie of undirected Toeplitz grph hve een invetigted in [] nd [5], while the directed ce w tudied in [6 8]. In [9], S. Mlik nd T. Zmfirecu trted the invetigtion of the Hmiltonin connectedne of directed Toeplitz grph. For the indirected ce, in [9] iti proven tht T n h; i i Hmiltonin connected onl for n D 3, while T n h; ; i i Hmiltonin connected for ll vlue of n nd. It will ecome cler tht, concerning k, the firt relevnt ce i k D 3. In thi pper, we re completing the picture of Hmiltonin connectedne of Toeplitz grph, more preciel of T n ht ;t i, T n h; 3; i nd T n h; 4; i. Let T e Toeplitz grph nd p; q e two vertice of T, uch tht p<q.b P p;q we men pth from p to p C contining ll vertice in fp; p C ; p C ;:::;q ; q ; qg, nd P q;p we men pth from q to q, contining the me vertice. The eitence of P p;q or P q;p i not gurnteed. We trt with few imple reult. Theorem. For n 3, T n ht ;t i i not Hmiltonin connected. Proof. Aume T D T n ht ;t i for n 3 i Hmiltonin connected. Then there eit Hmiltonin pth from t C to t C. But the pth from t C to t C contining i unique nd i of length. Thi led to contrdiction. Hence, T i not Hmiltonin connected. ut Theorem. The Toeplitz grph T n ht ;t ;t 3 ;:::; t k i i not Hmiltonin connected if t ;t ;t 3 ; :::; t k re ll odd. Proof. A iprtite grph i not Hmiltonin connected, nd if t ;t ;t 3 ; :::; t k re ll odd, then the grph T n ht ;t ;t 3 ;:::; t k i i iprtite. ut Corollr. T n h; 3; i i not Hmiltonin connected, when i odd. Theorem 3. If oth n nd t re odd, then T n h; t; n i i not Hmiltonin connected. Proof. Let, for t odd, T D T n h; t; n i, where n t C i n odd integer. Aume tht T i Hmiltonin connected, then there eit Hmiltonin pth H etween two even vertice nd of T. The pth H either contin the edge.; n/ or not. If H contin the edge.; n/, we cn contrct it to ingle verte, ecue oth vertice of the edge hve the me prit (oth re odd). After contrction, the reulting pth H 0 i of even order, nd the numer of even vertice i equl to the numer of odd vertice. But the end vertice of H 0 re even, which led to contrdiction. Net, we ume tht H doe not contin the edge.; n/. ButT without the edge.; n/ ecome T n h; ti, which i iprtite grph. Agin, H cnnot e Hmiltonin pth of T, nd thi complete the proof. ut Lemm. If n i even, then T n h; 3i dmit Hmiltonin pth from to nd, mmetr, nother one from n to n.
3 Hmiltonin Connectedne of Toeplitz Grph 37 Fig. n- n P, P,n Fig. Fig. 3 P +, P -,n Fig. 4 P -, P +,n Proof. See Fig., for Hmiltonin pth in T D T n h; 3i, from verte to verte, for even n 4. A imilr Hmiltonin pth from verte n to verte n eit in T, due to the mmetr of Toeplitz grph. Thi complete the proof. ut Lemm. Let p, q e two ditinct vertice of T n h; 3i. Ifq p i odd then pth P p;q nd P q;p eit in T n h; 3i. Proof. Appl Lemm to the ugrph of T n h; 3i pnned p; p C ;:::q. ut Theorem 4. T n h; 3; i i Hmiltonin connected for ll n C, if i n even integer. Proof. Let T D T n h; 3; i e the Toeplitz grph, where n C. Then, there eit pth P p;q nd P q;p in T, whenever q p i odd for p<q, Lemm. Now, uing uch pth of T, we prove tht for n two ditinct vertice nd of T, there eit Hmiltonin pth from to. Tke<. We plit our proof into two min ce: Ce. n i even. The following four uce rie: (i) i even, i odd. In thi ce, P ; nd P ;n eit in T, nd, with the help of thee two pth, we otin Hmiltonin pth P ; ;C ; C ;:::; ; ; P ;n in T from to ; ee Fig.. (ii) i odd, i even. If C, then poile Hmiltonin pth of T from to, PC; ;C ;:::;P ;n, i hown in Fig. 3. When D C, then for D or D n,weuethepthoflemm, nd for other vlue of, we conider the pth.;p C;n ;P ; ;/; ee Fig. 4.
4 38 M.F. Ndeem et l. P -, - P -,n Fig. 5 P +,+ + P -,n Fig. 6 Fig. 7 P,+ P +,n Fig. 8 P,+ + P,+ + P +,n Fig. 9 (iii) nd re even. If >, then Hmiltonin pth from to i.; ; P ; ; C ; :::; ; C ; C ; :::; ; P ;n / (ee Fig. 5). If, then we hve four uce to dicu:./ For > C, we conider Hmiltonin pth.; ; ;:::;;;P C;C ;C3;:::; ; P ;n / etween nd ; ee Fig. 6../ When D C n, then poile Hmiltonin pth joining nd i.; ; ;:::;;;C; P C;n ; P ;C ;CD/; ee Fig. 7..c/ If D C D n, then Hmiltonin pth from to i.; ; ;:::;3;;;P n;c /; ee Fig. 8..d/ Finll, for. A Hmiltonin pth joining nd i.; ; ;:::;;;P C;n ; ; ; 3; ; :::; C ; C ; P ;C / (ee Fig. 9). (iv) i odd, i odd. Thi ce i mmetric to ce (iii). (Denote verte i n C i.)
5 Hmiltonin Connectedne of Toeplitz Grph 39 P +,+ + P,n Fig. 0 P -,+ + P +,n Fig. P +,+ + P -,n Fig. + P +,+ P,n Fig. 3 Ce. n i odd. Agin, we conider the following uce:.i/ nd re of different prit. Firt, we ume tht <. Then Hmiltonin pth joining nd i.; ; ;:::; ; ; P ;n ; ;:::; C ; P C;C / (ee Fig. 0). If D, then Hmiltonin pth joining nd i.; ;:::;;;C; P ;C ; P C;n ;/(ee Fig. ). Net uppoe tht nd >C../ If i even, then Hmiltonin pth joining nd i.; ; ;:::;;;P C;C ;C3;:::; ; P ;n /; ee Fig.../ If i odd, then Hmiltonin pth.; ;:::;;P C;C ; C 3;:::; ; ; P ;n /, joining nd, i hown in Fig. 3. When < < nd D C, we conider Hmiltonin pth.; ;:::;4;;;3;P C;n ;P C;C / from to (ee Fig. 4).
6 40 M.F. Ndeem et l. P +, P +,n Fig. 4 P -+, -+ P -,n Fig. 5 P -+, -+ P +,n Fig. 6 P -+, P,n Fig. 7 If D nd D C, then Hmiltonin pth joining nd i. D ; ; P 3; ;P C;n ;/. Finll, here we conider the ce >../ If i even nd C, then Hmiltonin pth from to i.; ;:::; C; P C; ;C; C ;:::; ; P ;n /;ee Fig. 5. If D C, then Hmiltonin pth from to i.; P C;n ; ; ;:::; C ; P C; ;/(ee Fig. 6)../ If i odd, then Hmiltonin pth from to i.; P C; ; C ;:::; ; C ; C ;:::; ; ; P ;n /; ee Fig. 7. (ii) nd re even. The following uce rie: If D nd C,weue.;;C ;;:::;4;3;C 3; C ; C 5;:::; ; ; C ; P ;n /, the Hmiltonin pth etween nd (ee Fig. 8).
7 Hmiltonin Connectedne of Toeplitz Grph P,n Fig P,- P,n Fig. 9 P -+, +- P +-,- P,n Fig. 0 3 P -, + P +,n Fig. When 4 nd C, then Hmiltonin pth joining nd i.; C ;:::;C ; ; P ; ;C 3; C ; C 5;:::; ; ; P ;n /, hown in Fig. 9. If >,wehve.; C ; P C; ;P C; ;C 3; C ;:::; ; ; P ;n /, the Hmiltonin pth from to (ee Fig. 0). If <, the pth from to deired i.; ; 4; 5;:::;4;; C ; P C;n ;3;;5;6;:::; 3; ; C ; C ;:::; ; P ; /, when 0.mod 4/, nd.; ; 4; 5;:::;6;5;;3;P C;n ;C; ; 4; 7; 8; : : : ; 3; ; C; C;:::; ; P ; /, when.mod 4/; ee lo Fig..
8 4 M.F. Ndeem et l. Fig. P +, P -,n Fig Fig. 4 9 n n Fig n Fig. 6 (iii) nd re odd. In thi imple ce Hmiltonin pth from to i.p C; ; C ;:::; 3; ; P ;n / (ee Fig. ). Now the proof i complete. ut Lemm 3. For n D 5 nd ll n 7, T n h; 4i dmit Hmiltonin pth from to nd, mmetr, nother one from n to n. Proof. T n h; 4i i Hmiltonin for ll vlue of n ecept 6. See Fig. 3 for Hmiltonin ccle in T n h; 4i, when n f5; 7; 9g. Thee ccle re unique nd we ue them to find Hmiltonin pth from to in T n h; 4i. For n n 0.mod 3/, uitle pth i otined uing the Hmiltonin ccle in T 9 h; 4i; ee Fig. 4. To otin uch pth when n.mod 3/, we ue the Hmiltonin ccle found in T 7 h; 4i; ee Fig. 5. For n.mod 3/, the ccle T 5 h; 4i i emploed; ee Fig. 6. Now, ecue of the mmetr of the Toeplitz grph, we lo hve Hmiltonin pth from n to n. ut
9 Hmiltonin Connectedne of Toeplitz Grph 43 P -, - P,n P, + P +,n Fig P 4,n P 4,n c P 6,n Fig. 8 Lemm 4. Let p, q e two ditinct vertice of T n h; 4i. Ifq p ; 3; 5, then there eit pth P p;q nd P q;p in T n h; 4i. Proof. See Lemm 3. ut Theorem 5. T n h; 4; i i Hmiltonin connected for ll nd n 5. Proof. For n 5; let nd e ditinct vertice of the Toeplitz grph T D T n h; 4; i. Aume tht <. To prove the reult we how tht there eit Hmiltonin pth etween nd. Ce. D C. If D or n, we hve deired pth due to Lemm 3. When 5 n 5, then Hmiltonin pth etween nd i either.; P ; ;P ;n / or.p ; ;P C;n ;/(ee Fig. 7). When 4, ee Fig. 8 for Hmiltonin pth etween nd. For n 4 n, the deired Hmiltonin pth re mmetric to the pth for f; 3; 4g. Ce. If C, the following three uce rie: (i) n 5 nd f3;4;:::;n 5; n 3g. (ii) n 5 nd fn 4; n ; n ; ng. (iii) n 4. Suce (i). Let f3;4;:::;n 5; n 3g../ Firt, we ume the ce when f4;6;7;:::;n 5g.Now.P C; ;C ; C 3; :::; ; P ;n / i required Hmiltonin pth etween nd (ee Fig. 9)../ If D, then deired pth etween nd i hown in Fig. 30.
10 44 M.F. Ndeem et l. P +, + - P -,n Fig. 9 Fig. 30 P -, P 6,n 7 P 6,n c d P 9,n P 9,n e P -, Fig. 3 () A Hmiltonin pth etween nd 4. () A Hmiltonin pth etween nd 6. (c) A Hmiltonin pth etween nd 7. (d) A Hmiltonin pth etween nd 8. (e) A Hmiltonin pth etween nd, where P,n Fig. 3.c/ If D nd 5, then Hmiltonin pth etween nd different vlue of re hown in Fig. 3. When D nd D 5, to get deired pth, we ue the difference long with difference nd 4. See Fig. 3, for uch pth when f8; 9; 0; : : : ; n 6; n 4g. When D 5; 6; 7, ee Fig. 33.
11 Hmiltonin Connectedne of Toeplitz Grph P 7,n P 9,n c P 9,n Fig. 33 () D 5. () D 6. (c) D n-4 n n- n c d n- n n Fig. 34 () D n 5. () D n 3. (c) D n. (d) D n P 7,n P 7,n c P -,n Fig. 35 () A Hmiltonin pth etween 3 nd 5. () A Hmiltonin pth etween 3 nd 6. (c) A Hmiltonin pth etween 3 nd 7 And, for D n 5; n 3; n ; n, ee Fig. 34.d/ If D 3, then for Hmiltonin pth etween 3 nd, ee Fig. 35..e/ If D 5 nd 8, deired Hmiltonin pth i hown in Fig. 36. When D 8 nd n 5; 7, we ue the pth hown in Fig. 37. For n D 5 nd n D 7, ee Fig. 38 nd 39, repectivel.
12 46 M.F. Ndeem et l P 9,n P -,n Fig. 36 () A Hmiltonin pth etween 5 nd 7.() A Hmiltonin pth etween 5 nd,where P,n Fig c d e f g h i j Fig. 38 Hmiltonin pth etween 5 nd 8 for different vlue of,whenn D 5.() D 5.() D 6. (c) D 7. (d) D 8.(e) D 9. (f) D 0. (g) D. (h) D. (i) D 3. (j) D 4
13 Hmiltonin Connectedne of Toeplitz Grph c d e f g h i j k l Fig. 39 Hmiltonin pth etween 5 nd 8 for different vlue of,whenn D 7.() D 5.() D 6.(c) D 7.(d) D 8.(e) D 9.(f) D 0.(g) D.(h) D.(i) D 3.(j) D 4. (k) D 5. (l) D 6 Suce (ii). Thi uce i mmetricl to f; ; 3; 5g nd 6.Itw treted inide of.i/ ecept for the ce D n 4; n ; n ; n. To otin Hmiltonin pth from f; ; 3; 5g to fn 4; n ; n ; ng, we firt collect the four Hmiltonin pth in T 8 h; 4i from f; ; 3; 5g to 8; ee Fig. 40. Smmetricll, we hve pth in T n h; 4i from fn 4; n ; n ; ng to n 7, of verte et fn 7; n 6;:::;ng. Joining 8ton 7thedirectpth(8,9,...,n 7) give the deired Hmiltonin pth in T n h; 4i from to. Suce (iii). Thi uce i mmetricl with 5, treted inide of.i/. ut
14 48 M.F. Ndeem et l Fig. 40 To ee whether T n h; 4; i i Hmiltonin connected or not, for 6 n 4, ee the following tle: Hmiltonin connected when i T 6 h; 4; i T 7 h; 4; i T 8 h; 4; i 5, 7 T 9 h; 4; i 5, 8 T 0 h; 4; i 5, 6, 7, 9 T h; 4; i 5, 7, 8, 0 T h; 4; i 5, 6, 7, 8, 9, T 3 h; 4; i for ll T 4 h; 4; i 5, 6, 7, 8, 9, 0,, 3 Miing vlue for men tht the correponding Toeplitz grph i not Hmiltonin connected. Thi w verified uing computer. Reference. vn Dl, R., Tijen, G., Tuz, Z., vn der Veen, J.A.A., Zmfirecu, Ch., Zmfirecu, T.: Hmiltonin propertie of Toeplitz grph. Dicret. Mth. 59, 69 8 (996). Euler, R.: Chrcterizing iprtite Toeplitz grph. Theor. Comput. Sci. 63, (00) 3. Euler, R., LeVerge, H., Zmfirecu, T.: A chrcteriztion of infinite, iprtite Toeplitz grph. In: Tung-Hin, K. (ed.) Comintoric nd Grph Theor 95, Vol.. Acdemi Sinic, pp World Scientific, Singpore (995) 4. Euler, R., Zmfirecu, T.: On plnr Toeplitz grph. Grph Com. 9, 3 37 (03) 5. Heuerger, C.: On Hmiltonin Toeplitz grph. Dicret. Mth. 45, 07 5 (00) 6. Mlik, S.: Hmiltonin ccle in directed Toeplitz grph II. Ar Com. (to pper)
15 Hmiltonin Connectedne of Toeplitz Grph Mlik, S.: Hmiltonicit in directed Toeplitz grph of mimum (out or in) degree 4. Util. Mth. 89, (0) 8. Mlik, S., Qurehi, A.M.: Hmiltonin ccle in directed Toeplitz grph. Ar Com. 09, 5 56 (03) 9. Mlik, S., Zmfirecu, T.: Hmiltonin connectedne in directed Toeplitz grph. Bull. Mth. Soc. Sci. Mth. Roum. 53(0) No., (00)
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