Closed Monochromatic Bishops Tours

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1 Cos Monoromt Bsops Tours Jo DMo Dprtmnt o Mtmts n Sttsts Knnsw Stt Unvrsty, Knnsw, Gor, 0, USA mo@nnsw.u My, 00 Astrt In ss, t sop s unqu s t s o to sn oor on t n wt or. Ts ms os tour n w t sop vsts vry squr on t or xty on n rturns to ts strtn poston mposs. Wn n two sops, on n on wt, y vst vry squr (o tr rsptv oors) xty on n rturn to tr strtn postons? Su tour w os monoromt sop s tour. In ts ppr nssry n su nt ontons or t xstn o monoromt sop s tour or t rtnur mn or r provn. Furtrmor, monoromt nt s mov s n or t tr mnson ssor n os monoromt nt s tour s prov or t u o s. Introuton Puzzs on t ssor v on n stu y mtmtns. T survy pprs Comntor Proms on Cssors [] n Comntor Proms on Cssors II [] prov xnt ntroutons to t vrous typs o proms. T os nt s tour s on o ts mu stu n mous proms n t r o ssor puzzs. For w ors n nt y vst vry squr xty on n rturn to ts strtn poston? Swn ompty nswr ts quston or rtnur ors n. Torm An m n ssor wt m n s os nt s tour unss on or mor o t oown tr ontons o: () m n n r ot o; () m ; ; ; () m = n n ; ; : T unqu movmnt o t nt ms or ntrstn stuy w os tours or t n, qun n roo r trv to onstrut. T sop s oor

2 o n ry nnot tour vry squr on t or. Lt s n t quston sty. Wn n two sops, on n on wt, y vst vry squr (o tr rsptv oors) xty on n rturn to tr strtn postons? Su tour w os monoromt sops tour. Osony w w n monoromt tour tt vsts vry squr xty on ut os not rturn to ts strtn poston. Su tour s n opn monoromt sops tour. For ts ppr w w ssum t top t squr s wys n n nt m n ssor wt m n. T Cs o m = ; For t n ssor, t sops r un to m sn mov n no os monoromt sops tour xsts. For t n ssor t sops r to mov own t or ut r un to rturn wtout rptn squr. Tus, xpt or t ssor, no os monoromt sops tour xsts or m =. T Cs o m = No os monoromt sops tour xsts or m = n =. W tour o t wt squrs s poss (n sy), t squrs prov to troumrs. T our ornr squrs v ony two poss movs, on o w s to t ntr squr. For tour to xst ornr must pr or su y t ntr squr. Ts ors t ntr squr to vst mor tn on n no os monoromt sops tour xsts. Gvn t or ow, w n onstrut ny nt n 0 mo opn monoromt sops tour y pn ops o ts or s y s. Connt t pts y movn rom o t t or to o t rt or n rom o t t or to o t rt or. T sm movs w so onnt t wt pts. It s smp to onnt t two pts on t rtmost s y onstrutn t. Conntn t two wt pts rqurs t mor. Frst, t t n rt t n s. W r now t wt n opn monoromt sops tour wt ns on t tmost s o t or t n or ot t n wt tours.

3 B Wt Fur : T Bor to Crt t Opn Tour To rt os tour or ny n mo, prpn t pproprt r or or r n mo n oow t rt to os o t opn ns o t tour. x r x r x y s y s y z t z t z Fur : Bors to Prpn to Opn n Tour n mo Dt B Crt B 0 ; non y; y non t, t y, r y, r non y, t y, t, r; r n mo Dt Wt Crt Wt 0 non x; z; z; x non z, s z, s x, x non z, z, s, s, x, x E n or n xtn to m n or or m 0 mo. Not tt vry n or s rtmost s s nt. Crt two n ors p ntws. For ot n wt tours t t rtmost s n t top tours n t rtmost s n t ottom tours. Now rt t n s. Ts ys onstruton or os monoromt sops tours or t m n or or m 0 mo n n. T Cs o m =

4 0 0 m o 0 n p n 0 m Fur : Bs Css or m = Fur provs us wt v rnt os monoromt tours o nt n vryn wts. As wt t n ors, ts s ss n sy xtn to os monoromt sops tour or ny n or. T rt ow nts ow to us t os monoromt sops tour o t or to rt os monoromt sops tour or ny n or. n mo Dt B Crt B 0 n t n; n rt ; n t n; n rt ; n t n; n rt ; n t n; n rt ; n mo Dt Wt Crt Wt 0 n t n; n rt ; n t n; n rt ; n t n; n rt ; n t n; n rt ; Just t n ors, w n xtn ts n ors to ny m n or or m 0 mo n n xpt or t or. Ts nsstt t nuson o t s s. St two ors top to ottom s or. For t sop, t n t tmost s o t ottom or. In t top or, t n t tmost s t 0,, or (pnn upon t s s). Nxt onstrut t 0 n s (or otr s pnn upon t s s) to xtn t sops tour. For t wt sop, rmov t rom t tmost top or n t,, m n or o p (pnn upon t s s). Nxt onstrut t n s (or otr s pnn upon t s s) to xtn t wt sops tour. Ts onstruton wou not wor on t or s w wou n to t t tw.

5 Fronus Comntons I w n tt ts n n n tours to otr tn ny mn tour or m n onstrut rom t s ors s t Fronus numr (; ) = ( ) ( ) =. Fortunty, ts s sy to omps. P t pproprt numr o ops o t n tours ntws oow y t pproprt numr o n tours. On t rtmost s o t ors or t ottommost n or n upprmost n or oow t rt to omn t tours. Not tt t oor o t squrs or t n or w swt s on n mo. n mo Dt B Crt B 0 n n, n n ; n n, n n ; n n; n n ; n n; n n, n mo Dt Wt Crt Wt 0 n n, n n ; n n, n n ; n n; n n ; n n; n n ; T Cs o m = Unortunty, t Fronus numr (; ) = os notn to v os monoromt sops tours o t n or. So, on n, w prsnt s ss n mto to xtn tm to ny n or. Sn ts ours t t rt-most n o t n ors t os xst s outn wn rtn t os n ors:

6 0 0 n o 0 q 0 o m m p n Fur : Bs Css or m = n mo Dt B Crt B 0 t n; rt ; t n; rt ; t n; rt ; t n; rt ; n mo Dt Wt Crt Wt 0 t n; rt ; t n; rt ; t n; rt ; t n; rt ; Summry n Futur Wor T ov wor s to vry smr oon torm or sops s t on provn y Swn or nts. Torm An m n ssor wt m n s os monoromt sops tour unss on o t oown tr ontons o: () m = ; () m = n n = ; () m = n n = : Bsops on t m n ssor s not t ony sttn wr os monoromt tour ms sns. Gnrzn t squr ssor to

7 tr-mnson u so s ntury to os monoromt tour prom not ony or sops ut so or nts. T xtnson o t sops movmnt nto tr mnsons s sy to m n ry st oor o. T xtnson o nt s mov s not s ovous. On opton s to p t sm mov o nt w ms t nt trnt oors on mov. In 00, Qn n Wtns prov t xstn o os nt s tour o t sx xtror s o t rtnur prsm n [] w usn t nt. In t sm ppr, Qn n Wtns prov nt s tour o t u o s. In 00, DMo prov os nt s tour n t u o s n usn t nt xsts n ony n n vn []. Wy o ntr rts us t nt? Bus t nt s movs n t u o not trnt oor wt t nt! Just t sop, t nt s oor o. T nxt stp n ts rsr o monoromt tours s to trmn w us mt os monoromt tours wt t nt. As tsr, I v you wt n xmp o os monoromt nt s tour o t u o s, t smst u tt mts su tour. Gvn t symmtry o t u ony t tour s vn ow. Rott t u 0 rs or t wt tour Lv Lv Lv Lv Lv Lv Fur : A Cos Monoromt Knt s Tour n t Cu o S n =

8 Rrns [] J. DMo, W Cssors v Cos Knt s Tour wtn t Cu?, T Etron Journ o Comntors, Voum, (00) R [] G. H. Fr, S. M. Htnm, S. T. Htnm, A. A. MR, C. K. Ws, M. S. Joson, W. W. Mrtn, n W. D. Wy, Comntor Proms on Cssors: A Br Survy. Grp Tory, Comntors n Apptons () 0-. [] S. M. Htnm, S. T. Htnm, R. Rynos, Comntor Proms on Cssors: II, Domnton n Grps; Avn Tops, Mr Dr, In., Nw Yor,. [] A. J. Swn, W Rtnur Cssors v Knt s Tour? Mtmts Mzn : (Dmr ) -. [] Y. Qn n J.J. Wtns, Knt s Tours or Cus n Boxs, Conrssus Numrntum (00) - [] J. J. Wtns, Aross t Bor: T Mtmts o Cssor Proms, Prnton Unvrsty Prss, Prnton, 00. [] Wsstn, Er W. "Fronus Numr." From MtWor A Worm W Rsour. ttp://mtwor.worm.om/fronusnumr.tm