Research Article On the Genus of the Zero-Divisor Graph of Z n

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1 Intrntionl Journl o Comintoris, Artil ID 7, pgs Rsrh Artil On th Gnus o th Zro-Divisor Grph o Z n Huong Su 1 n Piling Li 2 1 Shool o Mthmtil Sins, Gungxi Thrs Eution Univrsity, Nnning, Gungxi 023, Chin 2 Shool o Mthmtil Sins, Gungxi Thrs Eution Univrsity, Nnning 023, Chin Corrsponn shoul rss to Huong Su; huongsu@sohu.om Riv 1 Frury 14; Apt July 14; Pulish 22 July 14 Ami Eitor: Lszlo A. Szkly Copyright 14 H. Su n P. Li. This is n opn ss rtil istriut unr th Crtiv Commons Attriution Lins, whih prmits unrstrit us, istriution, n rproution in ny mium, provi th originl work is proprly it. Lt R ommuttiv ring with intity. Th zro-ivisor grph o R, notγ(r), is th simpl grph whos vrtis r th nonzro zro-ivisors o R, n two istint vrtis x n y r link y n g i n only i xy=0. Th gnus o simpl grph G is th smllst intgr g suh tht G n m into n orintl sur S g. In this ppr, w trmin tht th gnusothzro-ivisorgrphoz n, th ring o intgrs moulo n,istwoorthr. 1. Introution This ppr onrns th zro-ivisor grphs o rings. For ommuttiv ring R, in simpl grph ll zro-ivisor grph, not y Γ(R), whos vrtis r th nonzro zroivisors o R, n two istint vrtis x n y r jnt i n only i xy = 0 in R. This inition ws irst introu y Bk in [1]. Howvr, h lt ll lmnts o R th vrtis o th grph n minly onsir th oloring o this grph. Hr our inition is th sm s in[2], whr som si proprtis o Γ(R) r stlish.th zro-ivisor grph,s wll s othr grphs o rings, is n tiv rsrh topi in th lst two s (s,.g., [3 11]). Lt us irst rll som n notions in grph thory. Lt G simpl grph, tht is, no loops n no multigs. Th gr o vrtx V G, not y g(v),isthnumr o gs o G inint with V. IV V(G), thng V is th sugrph o G otin y lting th vrtis in V n ll gs inint with thm. I V ={x V g(x) = 0 or 1}, thnwus G or th sugrph G V n ll it th rution o G.A omplt iprtit grph is iprtit grph (i.., st o grph vrtis ompos into two isjoint sts suh tht no two vrtis within th sm st r jnt) suh tht vry pir o vrtis in th two sts r jnt. Th omplt iprtit grph with prtitions o sizs m n n is not y K m,n.thomplt grph on n vrtis, not K n,isthgrphinwhihvrypiroistintvrtisis join y n g. A sur is si to o gnus g i it is topologilly homomorphi to sphr with g hnls. A grph G tht n rwn without rossings on ompt sur o gnus g, ut not on on o gnus g 1,isll grph o gnus g.wwritγ(g) orthgnusothgrphg. It is lr tht γ(g) = γ( G),whr G is th rution o G,n γ(h) γ(g) or ny sugrph H o G. Dtrmining th gnus o grph is on o th most unmntl prolms in topologil grph thory. It hs nshowntonp-ompltythomssnin[]. Svrl pprs ous on th gnr o zro-ivisor grphs. For instn, in [, 7, 13, 14], th uthors stui th plnr zroivisorgrphs(gnusqulsto0);wngtl.invstigtth gnus on zro-ivisor grphs in [11,, 1], rsptivly; n Bloomil n Wikhm trmin ll lol rings whos zro-ivisor grphs hv gnus two in [8]. In this ppr, w stuy th zro-ivisor grph o Z n, th ring o intgrs moulo n. In prtiulr, w trmin whn γ(γ(z n )) = 2 or 3. Hrwirstsummrizthrsultsoutthgnus o Γ(Z n ) rom [, Thorm.1()], [8, Thorm1],n[1, Stion ]. Thorm 1. Lt Γ(Z n ) not mpty. Thn th ollowing hol. (1) γ(γ(z n )) = 0 i n only i n {8,, 1,,,, 2p, 3p},whrp is prim. (2) γ(γ(z n )) = 1 i n only i n {,,,, 4}. (3) γ(γ(z p t)) = 2 inonlyip t =81.

2 2 Intrntionl Journl o Comintoris All rings onsir in this ppr will ommuttiv rings with intity. Lt n lmnt o ring R.Thnthprinipl il gnrt y is not y. For st A, A mns th orr o A. 2. Th Gnus o Γ(Z n ) Th ollowing two lmms r rquntly us in th proos o our min rsults. Lmm 2 ([17,Thorm.38]). γ(k n ) = {(1/)(n 3)(n 4)}, whr {x} is th lst intgr tht is grtr thn or qul to x. Lmm 3 ([17,Thorm.37]). γ(k m,n ) = {(1/4)(m 2)(n 2)}, whr{x} isthlstintgrthtisgrtrthnorqul to x. Lmm 4 ([17, Corollry.]). Suppos simpl grph G is onnt with V 3vrtis n gs. I G hs no tringls, thn γ(g) (/4) (V/2) + 1. Lmm. Lt G grph with vrtx st {u 1,u 2,u 3,u 4 ; V 1,...,V 7 ;w 1,w 2,w 3,w 4 } n th g st {(u i, V j ) 1 i 4, 1 j 7} {(V i,w j ) 1 i 3,1 j 3}.Thnγ(G) 4. Proo. Not tht thr r no tringls in G.AsG hs gs n vrtis, γ(g) 4 y Lmm 4. W irst onsir th s tht n hs only on prim ivisor. Thorm. Lt n=p t,whrp is prim n t 2.Thn γ(γ(z n )) = 3 i n only i n=4. Proo. ( ). Lt I= p t 1 {0}.Thn I = p 1.Asnytwo vrtis in I r jnt, thr xists omplt sugrph K p 1 in Γ(Z n ).Itollowsthtip 11,thnγ(Γ(Z n )) γ(γ(k )) = 4 y Lmm 2. Thror, p 7.Wpro with thr ss. Cs 1 (p = or 7). By Thorm 1, γ(γ(z 2)) = 0 n γ(γ(z 7 2)) = 1. So w n urthr ssum t 3.LtI = p t 2 p t 1 n J= p t 1 {0}.ThnI Jis mpty n I = p 2 p, J = p 1 4. As h vrtx o I is jnt to vry vrtx o J, thr xists omplt iprtit sugrph K 4, in Γ(Z n ), whih implis tht γ(γ(z n )) γ(γ(k 4, )) = y Lmm 3. Cs 2 (p = 3). From Thorm 1, whvγ(γ(z 3 2)) = 0, γ(γ(z 3 3)) = 0, nγ(γ(z 81 )) = 2. Sowmyssumt. Lt I = 3 t 3 3 t 2 n J = 3 t 2 {0}.ThnI J is mpty n I = =, J = = 8. Sin h vrtx o I is jnt to vry vrtx o J, Γ(Z n ) ontins omplt iprtit sugrph K 8,, whih implis tht γ(γ(z n )) γ(γ(k 8, )) = y Lmm 3. Cs 3 (p =2). By Thorm 1, γ(γ(z 2 t)) = 0 i t = 2, 3, 4,n γ(γ(z 2 )) = 1. I n =4, w n ssum t 7.WltI= 2 t 4 2 t 3, J= 2 t 3 {0}.Thn I = =8, J = = Figur 1: Th rution o Γ(Z 4 ). 0 8 Not tht h vrtx o I is jnt to h vrtx o J n I Jis mpty, so thr xists omplt iprtit sugrph K 7,8 in Γ(Z n ), whih implis tht γ(γ(z n )) γ(k 7,8 )=8y Lmm 3. Thror, n=4. ( ). For n=4,lti = 4 1 n J = 1 {0}. Thn I =, J = 3, n I Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K 3, in Γ(Z 4 ), whih implis tht γ(γ(z 4 )) γ(k 3, ) = 3 y Lmm 3. On th othr hn,wnmthrutionoγ(z 4 ) into S 3 s shown in Figur 1. Thror, γ(γ(z 4 )) = 3. This omplts our proo. W now onsir th s tht n hs xtly two prim ivisors. Thorm 7. Lt n=p s q t,whrp<qr prims n s, t 1. Thnγ(Γ(Z n )) = 2 i n only i n {,, 44, 0}, n γ(γ(z n )) = 3 i n only i n {, 2, 4}. Proo. W irst prov tht i 2 γ(γ(z n )) 3 thn n {,, 44, 0,, 2, 4}. W thn trmin γ(γ(z n )) or h n {,, 44, 0,, 2, 4}. W pro with our sstogtthrsult. Cs 1 (s =t=1). It is lr tht Γ(Z n ) is omplt iprtit grph K p 1,q 1.Inthiss,ip=n q 11,thnK 4, is sugrph o Γ(Z n );itollowsthtγ(γ(z n )) γ(k 4, )= 4 y Lmm 3. Ip = 2 or p = 3,yThorm 1, whv γ(γ(z n )) = 0.Sop=, q=7;thtis,n=. Cs 2 (s =1n t 2). I t 3,stI= pq t 2 pq t 1 n J = q 2 {0}.ThnI Jis mpty n h vrtx in I is jnt to h vrtx in J. Iq,thn I = q 2 q, J = pq t 2 1, whih implis tht Γ(Z n ) ontins omplt iprtit sugrph K,. Thror γ(γ(z n )) γ(k, )=y Lmm 3. Iq=3n t 4,

3 Intrntionl Journl o Comintoris 3 thn I = q 2 q =, J = pq t , whih implis tht thr xists omplt iprtit sugrph K,17 in Γ(Z n ). Thror γ(γ(z n )) γ(k,17 )=y Lmm 3. Hn,in this sitution, w hv n=2 3 3 =4. Consir now t=2;thtis,n=pq 2.LtI = q pq n J= pq {0}.ThnI Jis mpty n I = pq q, J = q 1. Noti tht h vrtx in I is jnt to h vrtx in J, so thr xists omplt iprtit sugrph K pq q,q 1 in Γ(Z n ).Iq=7,thnγ(Γ(Z n )) γ(k 7p 7, ).Iq=n p=3thn γ(γ(z n )) γ(k 4, )=4.ByThorm 1,wknow γ(γ(z )) = 0.Son=0. Cs 3 (s 2 n t = 1). Lt I = p s {0}n J = q {0}. Thn I = q 1, J = p s 1,nI Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K q 1,p s 1 in Γ(Z n ). Thror γ(γ(z n )) γ(k q 1,p s 1). Simply hking, w n s tht thr r only svn ss stisying th inqulity γ(k q 1,p s 1) 3;thtis,n {2 s 3,,,,, 44, 2}.By Thorm 1, whvγ(γ(z )) = 1 n γ(γ(z )) = 1. So n {2 s 3,,, 44, 2}. For n =, w st I = {8, 1,, }, J = {,,,,,, }, nk = {4,,, }. Nottht h vrtx in I is jnt to h vrtx in J,nthvrtis,, n r jnt to h vrtx in K,soyLmm, γ(γ(z )) 4. For th s n=2 s 3,yThorm 1,whvγ(Γ(Z )) = 0 n γ(γ(z )) = 1. Is = 4,thtis,n = 48,wst I = {8, 1,, }, J = {,,,,,, 42}, nk = {4,,, 44}. NotththvrtxinI is jnt to h vrtx in J, n th vrtis,, n r jnt to h vrtx in K, soylmm, γ(γ(z 48 )) 4. Is,wst I= 2 s 1 n J= {0}.Thn I = 4, J = 2 s 1 1 n I Jis mpty. Sin h vrtx in I is jnt to h vrtx in J,thrisompltiprtitsugrphK 4,2 s 1 1 in Γ(Z n ).Itthnollowsthtγ(Γ(Z n )) γ(k 4,2 s 1 1) 7 y Lmm 3. Cs 4 (s 2n t 2). W st I= p s 1 q t 1 {0}.Thn I = pq 1. Sin ny two vrtis in I r jnt, thr xists omplt sugrph K pq 1 in Γ(Z n ).Ipq 11, thn γ(γ(z n )) γ(k )=4y Lmm 2.Son=2 s 3 t or n=2 s t. I n=2 s 3 t,sti= 2 s {0} n J= 3 t {0};thn I = 3 t 1, J = 2 s 1,nI Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K 2 s 1,3 t 1 in Γ(Z n ).I 1 3or 2 3,thnγ(Γ(Z n )) 4.So s=2, t=2;thtisn=.in=2 s t,similrly,thrxists omplt iprtit sugrph K 2 s 1, t 1 in Γ(Z n ). Thror, γ(γ(z n )) γ(k 2 s 1, t 1) y Lmm 3. Nowwhvprovthti2 γ(γ(z n )) 3, thnn {,, 44, 0,, 2, 4}. In th ollowing, w trmin γ(γ(z n )) or h n {,, 44, 0,, 2, 4}. It is sy to s tht Γ(Z )=K 4,.Soγ(Γ(Z )) = 2 y Lmm 3. For n=,smntionincs4ov,γ(z ) ontins sugrphk 3,8.Soγ(Γ(Z )) γ(k 3,8 )=2. Γ(Z ) n m into S 2 s shown in Figur 2.Soγ(Γ(Z )) = Figur 2: Th rution o Γ(Z ). Sin th rutions o th grphs Γ(Z 44 ) n Γ(Z 2 ) r K 3, n K 3,,rsptivly,whvγ(Γ(Z 44 )) = γ(k 3, )=2 n γ(γ(z 2 )) = γ(k 3, )=3,rsptivly. For n = 0,smntioninCs2ov,whv γ(γ(z n )) γ(k 4, ) = 2 y Lmm 3. Wnmth rution o th grph Γ(Z 0 ) into S 2 s shown in Figur 3. Thus, γ(γ(z 0 )) = 2. For n=,wsti = {0} n J = {0}.Thn I = 4, J = 8,nI Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr is omplt iprtit sugrph K 4,8 in Γ(Z ).Itthnollowsthtγ(Γ(Z )) γ(k 4,8 )=3.Onth othr hn, w n m th rution o th grph Γ(Z ) into S 3 s shown in Figur 4.Thus,γ(Γ(Z )) = 3. For n=4,lti = n J= {0}.Thn I =, J =, ni Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K, in Γ(Z 4 ). Thror γ(γ(z 4 )) γ(k, )=3. On th othr hn, w n m th grph Γ(Z 4 ) into S 3 s shown in Figur. This omplts th proo. Th inl s is tht n hs mor thn two prim ivisors. Thorm 8. Lt n=p 1 1 p 2 2 p s s (s 3), whr p 1 <p 2 < < p s r prims. Thn γ(γ(z n )) = 2 i n only i n=, n γ(γ(z n )) = 3 i n only i n=42. Proo. Lt I = p 1 1 p 2 2 {0}n J = n/p 1 1 p 2 2 {0}. Thn I = p 3 3 p s s 1, J = p 1 1 p n I Jis mpty; morovr, vry vrtx in I is jnt to h vrtx in J. Thus, thr xists omplt iprtit sugrph K I, J in Γ(Z n ).Itthnollowsthtp 3 3 p s s 7s γ(γ(z n )) 3.So ithr n= or n= For th ormr s, st I= {0}n J = {0}; thn I = 4, J = n I Jis mpty. Not tht h vrtx in I is jnt to h vrtx in J, so thr xists

4 4 Intrntionl Journl o Comintoris Figur 3: Th rution o Γ(Z 0 ). Figur : Th rution o Γ(Z 4 ) Figur 4: Th rution o Γ(Z ). 3 Figur : Th rution o Γ(Z ). omplt iprtit sugrph K 4, in Γ(Z n ).I , thnγ(γ(z n )) 4 y Lmm 3. So 1 = 2 =1;thtis, n=. Lt I = {0} n J= {0}.Thn I =, J = 4,n I Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K 4, in Γ(Z ). It thn ollows tht γ(γ(z )) γ(k 4, )=2. On th othr hn,wnmγ(z ) into S 2 s shown in Figur, so γ(γ(z )) = 2. For th lttr s, with similr rgumnt ov, w hv n=42.lti = 7 {0} n J= {0}.Thn I =, J =, ni Jis mpty. Sin h vrtx in I is jnt to h vrtx in J, thr xists omplt iprtit sugrph K, in Γ(Z 42 ), whih implis tht γ(γ(z 42 )) γ(k, )=3. On th othr hn, w n m Γ(Z 42 ) into S 3 s shown in Figur 7,soγ(Γ(Z 42 )) = 3. This omplts our proo. Now w hv ompltly trmin whn γ(γ(z n )) = 2 or 3. W summriz th rsult y th ollowing thorm. Thorm. (1) γ(γ(z n )) = 2 i n only i n {,,, 44, 0, 81}. (2) γ(γ(z n )) = 3 i n only i n {42,, 2, 4, 4}.

5 Intrntionl Journl o Comintoris Conlit o Intrsts Figur 7: Th rution o Γ(Z 42 ). Th uthors lr tht thr is no onlit o intrsts rgring th pulition o this ppr. [8] N. Bloomil n C. Wikhm, Lol rings with gnus two zro ivisor grph, Communitions in Algr, vol. 38, no. 8, pp. 2 0,. [] Q. Liu n T. Wu, On zro-ivisor grphs whos ors ontin no rtngls, Algr Colloquium, vol.,no.4,pp.7 84, 11. [] J. Skowronk-Kziów, Som igrphs rising rom numr thory n rmrks on th zro-ivisor grph o th ring Z n, Inormtion Prossing Lttrs,vol.8,no.3,pp.1 1,08. [11] H.-J. Wng, Zro-ivisor grphs o gnus on, Journl o Algr,vol.4,no.2,pp. 78,0. [] C. Thomssn, Th grph gnus prolm is NP-omplt, Journl o Algorithms, vol., no. 4, pp. 8 7, 18. [13] N. O. Smith, Plnr zro-ivisor grphs, Intrntionl Journl o Commuttiv Rings,vol.2,pp.177 8,03. [14] N. O. Smith, Ininit plnr zro-ivisor grphs, Communitions in Algr,vol.,no.1,pp.171 0,07. [] H. Ching-Hsih, N. O. Smith, n H. Wng, Commuttiv rings with toroil zro-ivisor grphs, Houston Journl o Mthmtis,vol.,no.1,pp.1 31,. [1] C. Wikhm, Clssiition o rings with gnus on zroivisor grphs, Communitions in Algr, vol.,no.2,pp. 3, 08. [17] A. T. Whit, Grphs, Groups n Surs, North-Holln Mthmtis Stuis, North-Holln, Amstrm, Th Nthrlns, 184. Aknowlgmnts Th uthors thnk th nonymous rrs or thir vry rul ring o th ppr n or thir mny vlul ommntswhihimprovthppr.thisworkwssupporty th Ntionl Nturl Sin Fountion o Chin (1110) n th Gungxi Eution Committ Rsrh Fountion (LX14223). Rrns [1] I. Bk, Coloring o ommuttiv rings, Journl o Algr, vol. 11, no. 1, pp. 8 22, 188. [2] D. F. Anrson n P. S. Livingston, Th zro-ivisor grph o ommuttivring, Journl o Algr,vol.7,no.2,pp , 1. [3] D. F. Anrson, M. C. Axtll, n J. A. Stikls Jr., Zro-ivisor grphs, in ommuttiv rings, in Commuttiv Algr, Nothrin n Non-Nothrin Prsptivs,pp.23,Springr,Nw York, NY, USA, 11. [4] M. C. Axtll, N. Bth, n J. A. Stikls, Cut vrtis in zroivisor grphs o init ommuttiv rings, Communitions in Algr,vol.3,no.,pp.7,11. [] D. F. Anrson, A. Frzir, A. Luv, n P. S. Livingston, Th zro-ivisor grph o ommuttiv ring, II, Ltur Nots in Pur n Appli Mthmtis, vol. 2, pp. 1 72, 01. [] S. Akri, H. R. Mimni, n S. Yssmi, Whn zroivisor grph is plnr or omplt r-prtit grph, Journl o Algr,vol.0,no.1,pp.1 0,03. [7] R. Blsho n J. Chpmn, Plnr zro-ivisor grphs, Journl o Algr,vol.31,no.1,pp ,07.

6 Avns in Oprtions Rsrh Avns in Dision Sins Journl o Appli Mthmtis Algr Journl o Proility n Sttistis Th Sintii Worl Journl Intrntionl Journl o Dirntil Equtions Sumit your mnusripts t Intrntionl Journl o Avns in Comintoris Mthmtil Physis Journl o Complx Anlysis Intrntionl Journl o Mthmtis n Mthmtil Sins Mthmtil Prolms in Enginring Journl o Mthmtis Disrt Mthmtis Journl o Disrt Dynmis in Ntur n Soity Journl o Funtion Sps Astrt n Appli Anlysis Intrntionl Journl o Journl o Stohsti Anlysis Optimiztion