"""
cana is the list of Z(cprism)=n;
nonbr is the list of Z(cprism=n-1), connected graph, and neighborhood_condition fails;
MlZ is the list of graphs (up to 9 vertices) that Zhat(cprism)=n-1<n=Z(cprism);
"""
|
|
print "cana is the list of Z(cprism)=n;"
for n in range(5,9):
print "graphs on", n, "vertices with Z(cprism(g))=n";
for stg in cana[n]:
print stg;
show(Graph(stg),vertex_labels=False,vertex_size=50);
cana is the list of Z(cprism)=n; graphs on 5 vertices with Z(cprism(g))=n DLo  graphs on 6 vertices with Z(cprism(g))=n EIe_  ELv_  graphs on 7 vertices with Z(cprism(g))=n FHQSO  F@U^?  Fie`w  FHU^?  FsLZW  FLvvO  graphs on 8 vertices with Z(cprism(g))=n GICcSG  G?TclO  GIQ\T_  G]Q?x[  G?^VLo  G@Q?~? 
GKH?{W
 G_]?hK  G@LS^?  G@Q^Bo  G`CjcW  G@TctG  GIJK`s  GkGQ|W  GkGUXw  GHQmco 
GlQ@W{
 G_mqrK  G_]rSk  GF``Y[  G@U^fW  GJQ\CS  GLq@i[  G_|chk  GKT\LS  GKL\UK  G`ejrg  GBlleK  Gieb|w  GHU^No  GsT_x[ 
GkU`W{
 G]Ci[[ 
GkopW{
 G`YZtg  G@]vUg  G@]}fC  GHV^Ds 
Grj@y{
 GBq~V_ 
G]NB[{
Gs^`y{
 GJem^_  GJjLug  GK^d}w  G]Tk|[  Gj]\\k  GLvv^o 
cana is the list of Z(cprism)=n; graphs on 5 vertices with Z(cprism(g))=n DLo graphs on 6 vertices with Z(cprism(g))=n EIe_ ELv_ graphs on 7 vertices with Z(cprism(g))=n FHQSO F@U^? Fie`w FHU^? FsLZW FLvvO graphs on 8 vertices with Z(cprism(g))=n GICcSG G?TclO GIQ\T_ G]Q?x[ G?^VLo G@Q?~?
GKH?{W
G_]?hK G@LS^? G@Q^Bo G`CjcW G@TctG GIJK`s GkGQ|W GkGUXw GHQmco
GlQ@W{
G_mqrK G_]rSk GF``Y[ G@U^fW GJQ\CS GLq@i[ G_|chk GKT\LS GKL\UK G`ejrg GBlleK Gieb|w GHU^No GsT_x[
GkU`W{
G]Ci[[
GkopW{
G`YZtg G@]vUg G@]}fC GHV^Ds
Grj@y{
GBq~V_
G]NB[{
Gs^`y{
GJem^_ GJjLug GK^d}w G]Tk|[ Gj]\\k GLvv^o
|
print "nonbr is the list of Z(cprism=n-1), connected graph, and neighborhood_condition fails;"
for n in range(7,9):
print "graphs on", n, "vertices with Z(cprism(g))=n-1, connected, and neighborhood condition fails";
for stg in nonbrbk[n]:
print stg;
#show(Graph(stg),vertex_labels=False,vertex_size=50);
show_cprism(Graph(stg));
nonbr is the list of Z(cprism=n-1), connected graph, and neighborhood_condition fails; graphs on 7 vertices with Z(cprism(g))=n-1, connected, and neighborhood condition fails FK?}O  Fb]lg  graphs on 8 vertices with Z(cprism(g))=n-1, connected, and neighborhood condition fails GK_@iW  GIa?pK  G`GR[w  G@TfCw  G@TclO  GpQ@yw  GG_y~_  GK?}^o  GKLCG[  G`KakW  GXD[u[  G`YR|w  G`Lk~_  GoKyis  GI}TJk  GbjB|w  Go\|uk 
GlNB[{
 GJm}vK  Gbl|vK 
nonbr is the list of Z(cprism=n-1), connected graph, and neighborhood_condition fails; graphs on 7 vertices with Z(cprism(g))=n-1, connected, and neighborhood condition fails FK?}O Fb]lg graphs on 8 vertices with Z(cprism(g))=n-1, connected, and neighborhood condition fails GK_@iW GIa?pK G`GR[w G@TfCw G@TclO GpQ@yw GG_y~_ GK?}^o GKLCG[ G`KakW GXD[u[ G`YR|w G`Lk~_ GoKyis GI}TJk GbjB|w Go\|uk
GlNB[{
GJm}vK Gbl|vK
|
print "MlZ is the list of graphs (up to 9 vertices) that Zhat(cprism)=n-1<n=Z(cprism);"
print "graphs with Zhat(cprism(g))=n-1<n=Z(cprism);"
for stg in MlZ:
print stg;
show(Graph(stg),vertex_labels=False,vertex_size=50);
MlZ is the list of graphs (up to 9 vertices) that Zhat(cprism)=n-1<n=Z(cprism); graphs with Zhat(cprism(g))=n-1<n=Z(cprism); H?eBPhT  HWG]a}]  H]?XQ]M  H?_ys``  HJBK?sj  H`?nqy\  HLGW]VF  HA_x~ji  HK@~\rX 
H^QM?{n
 HK`hacN  H`HKXnW  H@H|uRP  H@~blnZ  Hk_zZiZ 
HoLR[~{
 H@]v^b\  H^qIz]v 
MlZ is the list of graphs (up to 9 vertices) that Zhat(cprism)=n-1<n=Z(cprism); graphs with Zhat(cprism(g))=n-1<n=Z(cprism); H?eBPhT HWG]a}] H]?XQ]M H?_ys`` HJBK?sj H`?nqy\ HLGW]VF HA_x~ji HK@~\rX
H^QM?{n
HK`hacN H`HKXnW H@H|uRP H@~blnZ Hk_zZiZ
HoLR[~{
H@]v^b\ H^qIz]v
|
cana={8: ['GICcSG', 'G?TclO', 'GIQ\\T_', 'G]Q?x[', 'G?^VLo', 'G@Q?~?',
'GKH?{W', 'G_]?hK', 'G@LS^?', 'G@Q^Bo', 'G`CjcW', 'G@TctG', 'GIJK`s',
'GkGQ|W', 'GkGUXw', 'GHQmco', 'GlQ@W{', 'G_mqrK', 'G_]rSk', 'GF``Y[',
'G@U^fW', 'GJQ\\CS', 'GLq@i[', 'G_|chk', 'GKT\\LS', 'GKL\\UK', 'G`ejrg',
'GBlleK', 'Gieb|w', 'GHU^No', 'GsT_x[', 'GkU`W{', 'G]Ci[[', 'GkopW{',
'G`YZtg', 'G@]vUg', 'G@]}fC', 'GHV^Ds', 'Grj@y{', 'GBq~V_', 'G]NB[{',
'Gs^`y{', 'GJem^_', 'GJjLug', 'GK^d}w', 'G]Tk|[', 'Gj]\\\\k', 'GLvv^o'],
9: ['H?A^@`H', 'H@QCBC[', 'H?CiSNo', 'H?CfaYL', 'HHOOSIA', 'H?YX?fB',
'H?DSRUs', 'Ho?y?SR', 'H@Ca[^o', 'H?Ci[^o', 'HK_a?[U', 'HK?yCCJ',
'H_?haiI', 'H@GZCfK', 'H@KcM`M', 'H@G[f@M', 'H?DSXr_', 'H?DXTbA',
'H?CZLbG', 'H@@[V?y', 'HAHCXzo', 'H@H?{^o', 'H?eBPhT', 'HBHF?yL',
'HHPDGyX', 'HGAuPpX', 'Ho@Xc\\y', 'HpOCzG\\', 'HI?X\\rE', 'Hk_@xhL',
'Ho@xcTZ', 'H?Tc^_}', 'HG?ytfK', 'H_@ZdU{', 'H?dP^Vs', 'H_@zSqr',
'Hk_BWwt', 'HG?ytjI', 'H?`pqvc', 'H?|_lfF', 'H?}@ilf', 'HHHIcqE',
'HsT_`CN', 'HIAm?sx', 'HIQs?sj', 'HLpC?[v', 'HKaBXpT', 'H_?zSvs',
'H?LRC^o', 'HGL?k^o', 'HGC[PNO', 'H`GRKO\\', 'H@YYCCr', 'H@HSV?]',
'H?YYd?r', 'H@HIcYQ', 'HWCGiqE', 'HGESPhI', 'HDoa?~e', 'HICcG~g',
'H@Qe@t[', 'HMGK?~e', 'HBOcG~g', 'HIALAs{', 'HIALAwy', 'HK?xuFL',
'HC`rPhJ', 'Hk`GhKz', 'HK@XtFL', 'HC`r`XJ', 'H@OZNE[', 'H?dRPnc',
'Ho?{qTt', 'H_]?j]u', 'H_]?l\\u', 'HKLTE?N', 'HIBL?sx', 'H@J_qeJ',
'H?YXaeb', 'HpOCxXT', 'Ho@xaUZ', 'HpODYg\\', 'HkQ?Xsv', 'HoAY`\\y',
'H?Lcmzi', 'H?LcnrM', 'H?URlrk', 'HO[P]jF', 'H?dTnP{', 'H@O\\]nw',
'H?L[vbe', 'H_kQZMt', 'H?UV`zl', 'H?dXvJq', 'HAG^fY]', 'H?TklVo',
'H?{qdNF', 'H]GQSGN', 'HKQAxqd', 'HKQEXot', 'HHHIcYQ', 'HIA\\Aoj',
'HIQT?wj', 'Hqgq?cN', 'HKXTC_N', 'HpOCxhL', 'HIa@zQT', 'Ho@xOvR',
'H?hy`fB', 'HoAyOtr', 'H?TktNo', 'H?{uBMV', 'H?LtfZY', 'H@I^RrT',
'H?LtnZY', 'H@I^Qzp', 'H@wmamV', 'H?Ltmrh', 'HkGQtGN', 'H@Qi|RP',
'HBaPY^g', 'HBEH]Vc', 'HGlQtIV', 'HWCs]p]', 'HAeRNO}', 'H?VLThq',
'H?U\\fPu', 'H@U[rJb', 'H?czubd', 'H?U\\v@t', 'H@I]Rvs', 'H@I]Rzq',
'HBEH]Za', 'H?pkres', 'H?d\\fPu', 'HWG]a}]', 'H@Jmctx', 'HlQ?XSV',
'HIkPIMF', 'H]Q?XSV', 'HF`aPKN', 'HGkqGnB', 'HYa@qgN', 'HqGTaWN',
'HYa?zOV', 'HsP_o[v', 'H?UZdfk', 'H?pkzap', 'H?EmZro', 'H?U^f?|',
'HFG^A]N', 'HFCi[^f', 'HYa@qWV', 'HkQ@pgN', 'H?|RKmf', 'H]?XQ]M',
'HL@KYuk', 'HK@YtUk', 'HiQ\\T_~', 'H_M\\Rhq', 'HdhI`[v', 'HIRstOz',
'H]Q@w~l', 'Hs_yr\\u', 'Hs@~Qwz', 'Hs_yr\\v', 'HJQ{tRB', 'HFye@lN',
'H]q?x\\v', 'HIQp~QZ', 'H?nRrjb', 'H?nbjfX', 'HIRpsuj', 'H?}ujrf',
'H?\\uvM{', 'H?zRtn{', 'Hs_yz\\v', 'H?CjdbK', 'H@HIcUS', 'H@EJL@X',
'H@HIfA]', 'H?_ys``', 'H_CbcW{', 'H@YQKOr', 'HHDD?~K', 'H@Rc@tY',
'H@Q?x^o', 'HsCa?|m', 'H@Q?zNw', 'H@Q?~F{', 'H@Q@~RS', 'H_Chb}}',
'HHHIcUS', 'H@TbCiI', 'HIGTGzH', 'HIA\\BOZ', 'HHQe?wZ', 'Hkgq?cN',
'HHRS@SZ', 'HheA?[v', 'HkU`?cN', 'HqGB[g\\', 'HqG@yiL', 'HkYP?cN',
'HGSrCUS', 'HHLAKYQ', 'HHQ[@dJ', 'H{H?ogN', 'HAb_pcj', 'HG_qW~o',
'H`Q@zQT', 'HK?q[^w', 'HGLRCYQ', 'H]CiCCN', 'HK?iO}M', 'H`QBhYX',
'HK_hiTT', 'HG_qp^S', 'H`Cha]M', 'H`Q@xZP', 'H`QF_wl', 'Hk`_Wkz',
'HG`Pp^S', 'Hq_WXLr', 'H`AXQni', 'HCCiZre', 'H`Agqni', 'H_[PK^u',
'HKO_}ym', 'HJBK?sj', 'HIBGpUr', 'H_]@_kf', 'HHQ[BCZ', 'Ho@X`^Y',
'HCDH^Hy', 'HqG@}G\\', 'HI?g}qe', 'H@Ci[^o', 'HoBXPdZ', 'HCOX^fk',
'HKE@yzk', 'H`?nqy\\', 'H`CeZy{', 'HpQ@YgZ', 'HKN?Yeb', 'HY`K`K^',
'HM`H_[f', 'HlQ?XKZ', 'HIoPG}e', 'HqIApgN', 'Hk`PO[v', 'H@YikrB',
'HKFPYqj', 'HGCytfK', 'H@Pk|RP', 'H?Szcng', 'HGkquIV', 'H@Tc^_}',
'HGC}mQx', 'HCLTMT{', 'H@dLJnw', 'Hja?Wsf', 'HFGiIUN', 'H@MImZa',
'HFQH[Xb', 'H@QJ[zo', 'H@LK^JQ', 'HCURNO}', 'HGFTYqx', 'HCURLT{',
'H@UKn`m', 'H@MeQ~s', 'H@MNA~s', 'H@EJ^JW', 'H@EI~Jg', 'HWMYKdZ',
'HGF]dOz', 'HWCyYeX', 'HsC_y\\l', 'HKLO\\VF', 'H@LIlNW', 'HLGW]VF',
'HGD\\mQx', 'HCURlPl', 'H@Lc^`]', 'HCLS^`m', 'HKKo]NJ', 'H@dN@~s',
'H@Q^Fp}', 'H@Q^Bt{', 'H@UNBm{', 'HK@Y|Ul', 'HK?y|ZJ', 'HITd\\a\\',
'H@YNdh\\', 'H`IlaxZ', 'H@MI~bc', 'H@Q}v@x', 'H_]k`lr', 'H@dN`zd',
'H@Rb|rX', 'HAJbuu{', 'HA_x~ji', 'H@MNazd', 'H@bpqtj', 'H@MfQzT',
'H@UNTht', 'H@Rbvq]', 'H@O{~Nw', 'H@R{vDz', 'HLoaGkN', 'HF_iPLF',
'HLq?pLF', 'HDoi`LF', 'HGsr?mF', 'HqI?z_N', 'HCcqZre', 'H@Lcmzi',
'H@LcnZY', 'H@LcnrM', 'H`N?ZER', 'HLqB?kN', 'HMi@QKV', 'HkGW]Cr',
'HbgOXNF', 'H@EJ]Zo', 'HGEXufk', 'HGE\\eT{', 'HK@~\\rX', 'H^QM?{n',
'HK@~T~]', 'HoDPS\\u', 'HpP_g]Z', 'HK_piTL', 'Hh_gg^R', 'HK_hiXR',
'HKLCI}m', 'HKLCM[}', 'HKDYtIj', 'HX?Y{rL', 'HBd@\\NK', 'H`GZSnK',
'HREKYXb', 'H`GUZY[', 'H@EZUVc', 'HLLCI]M', 'HQaIXlw', 'H`CZTNK',
'HXEKipN', 'HGfQtGz', 'HGfPmOz', 'HKfAhWz', 'HGfQhqj', 'HKDXuIj',
'HWEQ{Xx', 'H`CjdX]', 'HKQHXnW', 'HKG]UK{', 'HODZRe[', 'HoCqXxY',
'H@EiuVc', 'HB`I\\Us', 'HGfQlOz', 'HWK[qjF', 'H`Lcb]]', 'HBlCJ]u',
'H`Hkbu]', 'H`Hkdt]', 'H`Lca}m', 'HAddH~w', 'HGFXuaj', 'HBaYXV`',
'HGFYtaj', 'HXCYkVL', 'HGfQpij', 'HGd]Hex', 'HKCmTL[', 'HBci\\JB',
'HB`H\\NW', 'HLGY[ZB', 'HOLQrYU', 'HoDIhwy', 'HWMSYhZ', 'HkLCH[^',
'HBdA\\Mk', 'HKGYlVK', 'HXEKqhN', 'HKbHioz', 'HWDS]o}', 'H@Y]Bmy',
'H`DlBu]', 'HIJK`~y', 'H`DlDt]', 'H_hT`|{', 'HCTt@~i', 'H@Y]@~q',
'HkGRKoN', 'HxQ?okN', 'H@YYlRB', 'HKFQXqj', 'HGCytjI', 'H@Qj[rP',
'H?LZcza', 'HOCyrjI', 'HGlPuIV', 'HAM\\eDl', 'HGc}Iex', 'H@YYlRr',
'H@ebI~w', 'H@YkbnY', 'HkGWZER', 'HM_i_[f', 'HK`hacN', 'H`HKXnW',
'HKHG{^o', 'HWD{SdZ', 'H`CiZe{', 'H@LcmZI', 'HGePmT{', 'H`Cjc^{',
'H_KtI~w', 'H`Hkktx', 'HA[vNA\\', 'HA\\fLa\\', 'HBXVLQ\\', 'HHTcmU{',
'H@UfRi\\', 'HJDL^A\\', 'HBeqTTf', 'HBRl@vZ', 'H@QlrrS', 'H`KrSnK',
'H`MTItk', 'H@Mq]ra', 'H@Y\\RjQ', 'H@Lkuja', 'HII}Sdx', 'HChf`x\\',
'H@Vu@uj', 'HCUfPxt', 'H_lk`lr', 'H`lL@lV', 'HqYTA{}', 'HIiAx~s',
'HGfBh~w', 'HBQb^q]', 'HkGQ|^{', 'HJHL]a\\', 'H@hkzbP', 'H@LkvJQ',
'HIEb]u{', 'HxQSP|]', 'HBQb\\v[', 'HdOb{zl', 'HILd]a\\', 'HILT^A\\',
'H@YVayl', 'H`Ikitx', 'HHTcuM{', 'H`HtQyZ', 'H@hZtbD', 'H`Lciyi',
'H@iZRjQ', 'H@LkvbE', 'HIJ\\Sdx', 'HAefPxt', 'H@YV`zL', 'H`mJ@lV',
'HpUe@|]', 'HIEb^q]', 'HAqbh~w', 'HAebjzi', 'HkGUX~{', 'H@Y\\RfT',
'H@TenQ{', 'H@efaxl', 'H@TfeY{', 'H@efRh\\', 'HGLVczl', 'H@Td}jh',
'H@TevI{', 'H@Td}rd', 'HF@KZUk', 'HL?Y\\VK', 'HL?g}ZI', 'HK?ytVK',
'HXQIko^', 'H_Lkres', 'H@YkrjQ', 'HbhK`kn', 'H_mahlx', 'HAev@xj',
'H_]d`l\\', 'HhHKsk^', 'H@Tc|Zo', 'HBRkv?z', 'HHQmkpx', 'H@Y^Dd\\',
'HAdlbel', 'HIL_{^o', 'HHQm`xZ', 'HkU`a}m', 'HlQAx]|', 'HkU`e[}',
'HAcz\\jb', 'HHQ}tRP', 'HlUe@[^', 'HIIqtvM', 'HlQA|W~', 'HIIq|rJ',
'HkBytSz', 'HG~@jeV', 'HFf@ZTf', 'HLue@lN', 'HGnAh~q', 'HHjAh~Y',
'HmMe?{n', 'HFqBjWn', 'H_[tI}u', 'HAwsj]u', 'HGnB`~U', 'HGvBhmx',
'HIIqvU]', 'HlQB[w~', 'H_]tI~y', 'HkBlqs|', 'H_|qlUv', 'H_]zSnr',
'H@X^S~s', 'Hk@|qyj', 'H_|rKmz', 'HHS^NA\\', 'H@X\\]zq', 'H@|RKmf',
'HHRa|qZ', 'HBRb\\qZ', 'HHQ}vAX', 'Hk_yzel', 'HBY~EaJ', 'H]o\\@lN',
'HFj@Y\\r', 'HGupjfJ', 'HGnPqnb', 'HInAhuf', 'HkOpX}]', 'HJQ|UaJ',
'H]N?\\LZ', 'Hs\\PW~f', 'HCTztVd', 'H@TmvM{', 'HkBmpwz', 'HCLZ~Rd',
'HGUrnQZ', 'HGUrlrJ', 'HAf~Buz', 'H@L[~ji', 'HF_i^L}', 'HAcx~Fd',
'HFy@i\\f', 'H_[tK|u', 'HHQysvb', 'HG~Ahmr', 'HGuhjfR', 'Hjee?{n',
'HAwp{nd', 'HGub`~U', 'H_\\tK~y', 'HqAzuS|', 'H@U^vH|', 'H@U^fzm',
'HAc~vjl', 'H]_~?|N', 'HFOz[^h', 'Hk`xucn', 'Hk`zdS^', 'HFHJ[~k',
'H{_yplN', 'Hs\\TG|n', 'HAfvRu|', 'H@U^vjk', 'H@H[~Ro', 'HBRlU_z',
'HHRc{px', 'HAff@s|', 'H_ld`l\\', 'HIW^La\\', 'H]LCJ]]', 'HKbvPxZ',
'H{SoZUV', 'H^`KP\\V', 'H]hK`\\V', 'HJQstVK', 'HNe@Y\\f', 'H}LCG{n',
'HImAh~e', 'HJCmUM{', 'HlLCI}m', 'HlLCM[}', 'HlLCJ]]', 'H@H|uRP',
'HAdrTii', 'HILVLQ\\', 'HoCyv^u', 'HJRstQJ', 'HKfBZgz', 'HF@KZYi',
'HRAKZXY', 'HL@KYyi', 'HR?kY^W', 'HR?Y\\ZI', 'HR?{YVH', 'HK?y|RH',
'HL@KZU[', 'HK@itU[', 'HXQIsg^', 'H@IZvRS', 'HBZsUCz', 'HBRleOz',
'HJJ[UCz', 'HJCi[^o', 'HkYPa}m', 'HkYPe[}', 'HAdrTek', 'HBZcmOz',
'HHZS]_z', 'HjQ\\CS~', 'H`IsrXZ', 'HISvLQ\\', 'H_Mczhw', 'H@YRlrK',
'H`GZ[zo', 'H@TktNo', 'HJRLSgz', 'HHQu[px', 'HsHHqk^', 'H_LtbYZ',
'H_Lcziw', 'H_Lkzap', 'HCTdlXw', 'H`MTIxi', 'H`Ksi^g', 'H@Q\\Zro',
'HblCHkn', 'HHZSuGz', 'HHRsuOz', 'HJRcsWz', 'HHTVLQ\\', 'H_Mtaxj',
'H_Lkriq', 'H`LcjYY', 'H@YZdfK', 'HJRkcSz', 'HbhSPkn', 'H{L?i}m',
'HMiAx^t', 'H{L?h~M', 'H`O|Yyr', 'H_M\\Zhr', 'HF_zI^j', 'H_Lkzir',
'HqA|rXZ', 'HK`ztjJ', 'HqBlqwz', 'HK`~PzR', 'Hk@|rYZ', 'HCSz^Nw',
'HkBzSsz', 'H\\o]@lN', 'HIq`h~Y', 'HGubpnT', 'Hye@W|v', 'HGubriV',
'HF`j`\\N', 'Hmeb?{n', 'HIv@pmf', 'HNqLA[v', 'Hn`L?{n', 'HByAh~e',
'HlYU@[^', 'HHQytfJ', 'HIu@h~e', 'Hk@|rU\\', 'HKbzauj', 'HCLZ^Zq',
'HCTjt^s', 'H_}qhvf', 'HG}Pinb', 'HA{slLf', 'HG{qknf', 'HA}`jfF',
'H@{rKnF', 'HGvBpmt', 'HIIquum', 'HIy@i}u', 'H_|ch~u', 'H_}`i~u',
'H{`qp[^', 'HXL[u^U', 'HXNMa}^', 'H@~blnZ', 'HXL[}^V', 'H]`l_|N',
'HFHjs^L', 'H{RPo{n', 'Hkbhqsv', 'HqIzeS^', 'Hkayrcn', 'H}_ox\\N',
'HFOj\\^[', 'HkazRc^', 'HLOx}VL', 'Hswqg|n', 'H@]}vFt', 'HAd|vfk',
'HAd|~bh', 'H@nvMpz', 'H@]~Ujr', 'HUOix~k', 'HFHJ|ZL', 'Hkbhrc^',
'HqI~As^', 'H@]vUn{', 'H@]u~Rt', 'HAd|vji', 'H@n}fDz', 'HBZvTzZ',
'H]hK_|f', 'HNq@h\\N', 'HJQt[rH', 'HbWhk]V', 'HBzAhmj', 'H}_XPlN',
'Hiebc[~', 'Hk_zZiZ', 'HoLR[~{', 'HkUhW~r', 'HKNEX~{', 'HJYucYJ',
'HrN?\\LZ', 'HtoZ@lN', 'HbgpYmN', 'HJYuSiJ', 'HKNB[~{', 'HBYleVS',
'HtoZ@\\V', 'HLu@i\\f', 'H]@Z[ul', 'HKTX|Nh', 'Hk^@W}v', 'HkoxW~r',
'HKNA|^{', 'HB]lmVT', 'H]opYmN', 'H]_}Qkn', 'HlU`W~N', 'HkT\\LS~',
'HAd|vre', 'HXL[}jJ', 'HkNe`[^', 'HropYmN', 'HUHjYyZ', 'HLrcplN',
'H]Ogz]]', 'H]`kqkn', 'HUOjX~[', 'HlYPW~N', 'HKT\\|rd', 'HXUY{zb',
'HXeYzrF', 'HKT\\\\vs', 'H`h\\vh}', 'H`]zSnr', 'HHUu\\v[', 'Hdejjpn',
'HxSysmN', 'HYd[r]u', 'H`Xzsyr', 'HXeYrnM', 'H`mi~`v', 'HXM]Q~V',
'H@]v^b\\', 'HXL[u^V', 'H`Lt^r]', 'HBllfN]', 'HtopYlN', 'H]`\\Qkn',
'Hy`[pkn', 'HtoXYlf', 'HyaZO{v', 'HpP|Uc^', 'HLPX}Yj', 'HLPX{^h',
'HBYu\\v[', 'HBllenm', 'Hiefzy|', 'HI}srnf', 'H`Ljk~w', 'H@]}ujb',
'H@\\tvN[', 'HHV{vDz', 'HHU^Nv{', 'HHV^Lvx', 'HHV~DvZ', 'HLr@htN',
'HkSpW}f', 'Hk_xzjJ', 'HsXSx\\|', 'HpPr[yZ', 'HkUax]|', 'HrciW~f',
'HpQm`|]', 'HqHuXyZ', 'HkNAx]|', 'H]?z[zJ', 'HpQmhxZ', 'H]@Z[yj',
'H]@j[u\\', 'HpR`yyZ', 'HpLIk|}', 'HpKyYlZ', 'HHNU]u{', 'HdUljpn',
'H`\\Z\\ir', 'HHUu\\zY', 'HKL\\unk', 'H[eYr\\u', 'HDljfN]', 'H@lvnR\\',
'H@]~fF\\', 'H`hy|rr', 'HYU[zij', 'HKS|}rd', 'H@vZtjb', 'HB]lmZR',
'HKcy~ji', 'HlogzMV', 'HLqm`lN', 'HsDjXxZ', 'H]`lQk^', 'H]CizYN',
'HUHip~M', 'HUOjZy]', 'HKS|]zq', 'HKS|]vs', 'H@t}lrb', 'H@\\t}rd',
'HDhv]p|', 'H`Yy|rr', 'HBllmrf', 'HHV^Fu}', 'HNOi{]l', 'HBZMlyy',
'HB]lenm', 'HJ]lmVT', 'HI~crmv', 'HimuvG~', 'HJYu\\zZ', 'H}e`y|n',
'Hrqb}w~', 'HJZutU\\', 'HBY}vfk', 'HFyuRln', 'HI~cjmz', 'HlUf]w~',
'HWeZzrT', 'HWUZ|rT', 'HWUZlv[', 'HWL\\un[', 'HWL\\}rT', 'HWMY}vs',
'HjY\\c[~', 'HJY}tfL', 'Hl]emW~', 'HHV~fQZ', 'HjemnO~', 'HlU`z]^',
'HHVvTv\\', 'HFxr\\nN', 'HzqTY{~', 'Hzj@y}^', 'Himp}vf', 'HFY}~Zj',
'HFjJ~^{', 'HwEuP|]', 'HWeZjv[', 'Hj]KlK~', 'HJZ|eUZ', 'HjeneW~',
'HBY}~Rp', 'Hlle]g~', 'HF}eJ\\v', 'HWd[zzq', 'HWUZtn[', 'HJZutYZ',
'HJY}tjJ', 'Hl]uUK~', 'HJ]lmZR', 'HjemvG~', 'H^qIz]v', 'H`vj|zr',
'Htozi|n', 'Hjeh}vf', 'H^Tk|^N', 'Hlpkz}}', 'Hl\\|lvN', 'HLvv^zy'], 1:[], 2:[], 3:[], 4:[], 5:
['DLo'], 6: ['EIe_', 'ELv_'], 7: ['FHQSO', 'F@U^?', 'Fie`w', 'FHU^?',
'FsLZW', 'FLvvO']}
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nonbrbk={1: ['@'], 2: [], 3: [], 4: [], 5: [], 6: [], 7: ['FK?}O', 'Fb]lg'], 8:
['GK_@iW', 'GIa?pK', 'G`GR[w', 'G@TfCw', 'G@TclO', 'GpQ@yw', 'GG_y~_',
'GK?}^o', 'GKLCG[', 'G`KakW', 'GXD[u[', 'G`YR|w', 'G`Lk~_', 'GoKyis',
'GI}TJk', 'GbjB|w', 'Go\\|uk', 'GlNB[{', 'GJm}vK', 'Gbl|vK'], 9:
['HIa??se', 'H@Q??~a', 'HKH?c?N', 'HK?@OnK', 'H`Q?@SU', 'H_?Gjqe',
'HK_a?gJ', 'HKQ?OKr', 'H_?zC_J', 'HoCG`LF', 'HWCOQmM', 'HGCGlrE',
'HKGWONB', 'HK@KPCT', 'HKO_gUL', 'H_KQOMd', 'H_KPA]u', 'HGEOQmi',
'HK_OPlM', 'H@IJK`X', 'H_Chd`M', 'HGEK`hI', 'H?Cq\\RO', 'H?hqS_r',
'H@@ku?x', 'HKE@_\\L', 'HHPCG}w', 'H`@?x]w', 'H`@@wyh', 'HICd?~K',
'HIAkAsy', 'HU`@?|m', 'HIA?|pm', 'H@B@zQX', 'HABL@tw', 'H?h@jqU',
'HGAHud{', 'HcG?~Xu', 'H`_?zze', 'H`ChCdM', 'H`G?|hM', 'HWCWaUF',
'H@Cc^@[', 'H`KPCLM', 'H`G@kxM', 'H_?X\\pu', 'HK_Biyk', 'H_@|Ovp',
'H?_vaxl', 'H_@tO~w', 'H_?^ril', 'HIHCWyp', 'H?`ZPno', 'H?`y`va',
'H`G_qmM', 'HK_i?cF', 'HKO_sGN', 'H`GZC_N', 'HWCGiqF', 'HGE[@dI',
'HKGQKSN', 'HWCWaUE', 'HGCShZG', 'H`KcI_N', 'HKQ?Zmy', 'HKQ?^c}',
'HF?iS\\m', 'HsC_Y|m', 'H?UV@~{', 'H`H_o]R', 'H`H@g]X', 'H`H?w]p',
'HWCs}P\\', 'HsCgYtf', 'H@JURyy', 'HYa?yof', 'HI_PY]s', 'HGsq`MF',
'HaOx|Yr', 'Hs?yr~m', 'H`GoSdM', 'H`GGlL]', 'H`?gr]]', 'HIQT?sl',
'H`C_ZUU', 'Ho?xa^Y', 'H?hpavE', 'HC`j`hJ', 'HOSP^jM', 'HHQSPdL',
'HLQ@GsN', 'HK_OXlM', 'HKGW^NY', 'HKGW^fM', 'H`H@WyR', 'H`L?Wmb',
'H`Ggq]U', 'H_GXi]w', 'H_Chlp]', 'H`GXb]]', 'HoEYPLr', 'HY`HGmZ',
'HK?iuW}', 'HMCIXYf', 'HK`TG\\x', 'HK?Y\\T{', 'HK`PW^p', 'H_CXthm',
'HpC_]\\]', 'HKS_\\nM', 'HODJ`~[', 'H`CaX}{', 'HKL?\\nM', 'HGEUP~{',
'H`CcZ~{', 'Hkop?cN', 'HwC@yiL', 'HHDCWz`', 'HBe@PLF', 'H`C_\\dM',
'H`CazY{', 'HK?X}rk', 'HGEUtX{', 'H`Cczzk', 'H`Caz]{', 'H_G^kxx',
'H`GR[~{', 'HGEQ~Q{', 'HGEP}rk', 'HODMpxt', 'HGC\\ujk', 'HOC]rXt',
'H_C^Zqt', 'HL?gyUL', 'HqC`sXF', 'HKBYpqj', 'HBci\\FD', 'HOCZjrK',
'H`@ZTo}', 'HG_}iqx', 'H@TfC~{', 'H`A\\Rxy', 'H@dlA~q', 'H@Y\\BnY',
'HwCSqWf', 'H@MImVc', 'HWCyqiJ', 'HoCqXt[', 'HOLIriU', 'HLG[YNH',
'H`@zSoz', 'HGbYt_z', 'HOCyZfW', 'HYOP[yN', 'HG`\\iqx', 'H`@kp~w',
'HA`l`~w', 'H`@kru{', 'HBhKb]u', 'H`Alqt|', 'HAb`xvx', 'H@RP|Vx',
'HGu@j~u', 'H@Rcr~y', 'H`@kr}}', 'HG[qk^o', 'HHTd]a\\', 'HhOKh}]',
'HHTc]e{', 'H@ejRjQ', 'H@TklVo', 'HAavPxx', 'H`@|PvX', 'HpQ@yw^',
'HpQBxz\\', 'H_DtZyy', 'H_Eljxy', 'HIIa}u{', 'HIIa~q]', 'HGmBjzU',
'H@O{~Zq', 'H_C|jvk', 'HG`^lp|', 'H`Ayvvm', 'Hb?\\T\\]', 'H@MNIzp',
'H`AhzpZ', 'H`Axqtj', 'H`WKll]', 'HHRA|u{', 'HG_y~ru', 'HG`X~jy',
'Hb?Z\\U\\', 'H_I|atx', 'H@Yfay\\', 'H`@kzqx', 'HBlKbMf', 'H`WKjm]',
'HGuBh~s', 'H_Czl^w', 'HA_z\\nw', 'HK?}^v{', 'HKBxvVZ', 'HIgPG~E',
'HGcrOnD', 'HF?hYYJ', 'HM`HPKV', 'HK`i`SV', 'Hb?XXUL', 'H`?Xq]k',
'H`?hYyY', 'HKGZKvK', 'HGeIn_}', 'HGeHmd{', 'H`KcY~s', 'H`KTI~k',
'HGFruqj', 'HKB~Uoz', 'HK@~Uu{', 'HK@|vV\\', 'H]YU@[^', 'HGuJhnp',
'HG}BhnT', 'H{E@yxn', 'H[@HWt\\', 'HkGOY}m', 'HWDSP~]', 'HkGO][}',
'HoCqP|]', 'H`HSR}}', 'H`KoZUU', 'H`KpQmM', 'HGEYvam', 'HLECY\\k',
'HOLIjqU', 'HWEYlPZ', 'HWCZkrL', 'HBRL@~y', 'H`HM`}{', 'H`ai`|y',
'H@hUH~w', 'HHQS^p}', 'H`YUB}}', 'HWC[mpm', 'HXC[mPN', 'HOCyjrI',
'HWCYjqM', 'HODJjq[', 'H`KRLX]', 'HWDS{Xx', 'H`HSZyy', 'H`Qk`|y',
'H`HSX~w', 'HaddB}}', 'HqCdOxF', 'HI_Ox^c', 'HK?gyuk', 'HGCys^o',
'HB_i\\VS', 'H`HIlo}', 'H@eeRl{', 'HGLVC~{', 'H`IKjxy', 'HGLVDz]',
'HkGPWzB', 'HM``O[V', 'Hk?hWs\\', 'H`?gzYY', 'H`?XY]w', 'HA`ljqw',
'HIJksdx', 'HGfBjyy', 'HqGTyzl', 'HwMUA{}', 'HkGR[~{', 'H]?gyYJ',
'H_DtZqw', 'H@Lk~BP', 'H_S|\\dt', 'Hk@\\R}}', 'Hk@\\Zyz', 'H_Y^b}}',
'HDSz]Zb', 'H_yriu|', 'HAe~B~y', 'H@NebYZ', 'HxQ@w~\\', 'HGhX}jr',
'HFKi[^f', 'HFHKZ~m', 'HkBHr}}', 'HCL^B~m', 'HDuBjXf', 'HCU~Rzr',
'H@U}~fl', 'HAd|vfl', 'H{@ixu\\', 'HCTjtze', 'HKb^bWz', 'HGmRrjF',
'HGuRpnd', 'HGuRhnh', 'HHZSmOz', 'HjYKcK~', 'H`MTIxj', 'H}GOZ]]',
'H[dBW~t', 'H`?\\Yzg', 'H`?]\\Xw', 'HNAKXXJ', 'H`KdYzS', 'H`LcjYZ',
'H`lDHl\\', 'H`]DHl\\', 'H[oPy^t', 'HmI@w~l', 'Ha_xzij', 'H{DAx]|',
'HmIBW}|', 'H_Sz\\et', 'HE`j`~m', 'Hq_g~l}', 'Hq@Xv]}', 'HK`\\Vl}',
'HHQq~QZ', 'HGiZrjR', 'HGyR`~U', 'HK`|rVT', 'H_}qpnf', 'H_^ch~y',
'HMeBjWn', 'HDfBZhj', 'H_^`k~y', 'Hk`zcsn', 'H@NuBUZ', 'H@]}~Zr',
'H@^d~jZ', 'HBY}~fl', 'HXDKY}]', 'HaWp{ml', 'HVpL@\\V', 'HSXirm]',
'Ho[ytNV', 'HC]ub~m', 'HXD[u~m', 'HKT\\T~u', 'H}GWZUV', 'H{cZ@lN',
'H`L[zij', 'H`M[zXr', 'HoTXrm}', 'HpQuXt\\', 'HSXip~U', 'Hk]Ox^f',
'HoSp}^{', 'HkYXW~r', 'HWeYr~u', 'HKc}b~m', 'HKS}T~u', 'HWUUzy|',
'HBjB|~{', 'H`YR|~{', 'HIeb|~{', 'HgK{{lx', 'HKeuZxy', 'HTdje\\}',
'HLYZlrN', 'HJVdU}}', 'HWM]rzV', 'HIncj~y', 'H{D[rKn', 'HtOXY|m',
'HVOjYyN', 'HUOizym', 'H`L\\nZy', 'Hie`z}}', 'H`YZ|zr', 'HdUdzxn',
'Ho^@k|}', 'H{@Yxyj', 'HWM]b~]', 'HIUle}}', 'H{Kgy^V', 'Holak|}',
'HXDZszN', 'H{ciW|v', 'HWd\\b~]', 'HHUuT~]', 'HKL^D~]', 'H`Tlb}}',
'HKNE|x{', 'HKMm}hx', 'H`llmpv', 'H`mjuhv', 'HKNL}hx', 'H`MzVf]',
'HJjCz~{', 'HIutJ~y', 'HI}TJ~u', 'HDlvMT|', 'HBlt]rf', 'HJYuT~]',
'HKc}Zzq', 'HgK{i}y', 'H`LZlvk', 'HJZMlyz', 'HI}kjnr', 'HbjFzy|',
'Hyeb}w~', 'HI}sjnj', 'HJVd]yz', 'HIh}|zr', 'HDpz~jj', 'HwUX{xr',
'HWUR|z[', 'H{LT}xn', 'HiM\\\\|}', 'HFhj~jN', 'HBu~V^u', 'HB]}~jj',
'HpP|eS^', 'HlYV]w~', 'HJZ]tYr', 'H}YQ|[~', 'HJ]}v^u', 'HL]}^NZ',
'HNjL}xn', 'HpTz|zZ', 'Hb\\|\\mz', 'HL]~vjN', 'Hbl|vnm', 'Hb]|~Zr']}
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##MlZ is the list of graphs with M(cprism)=n-1=Zhat<Z=n
MlZ=["H?eBPhT","HWG]a}]","H]?XQ]M","H?_ys``","HJBK?sj","H`?nqy\\","HLGW]VF","HA_x~ji","HK@~\\rX","H^QM?{n","HK`hacN","H`HKXnW","H@H|uRP","H@~blnZ","Hk_zZiZ","HoLR[~{","H@]v^b\\","H^qIz]v"]
##j and k are graphs with 9 vertices M(cprism)=8, but Z(cprism)=8 also.
j="H@HAC?^";
k="H@HAC?^";
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### Below here are just codes -----------------------
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URL='http://github.com/jasongrout/minimum_rank/raw/minimum_rank_1_1_3/'
files=['Zq_c.pyx','Zq.py','zero_forcing_64.pyx','zero_forcing_wavefront.pyx','minrank.py', 'inertia.py']
for f in files:
load(URL+f)
load('https://raw.githubusercontent.com/jephianlin/minimum_rank_aux/master/mu_dic.py');
load("https://raw.githubusercontent.com/jephianlin/minimum_rank_aux/master/deltaCeiling.py");
load("https://raw.githubusercontent.com/jephianlin/minimum_rank_aux/master/oc_diag_analysis.sage");
load("https://raw.githubusercontent.com/jephianlin/minimum_rank_aux/master/xi_dict.py");
Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_qygsEw.pyx..\ . Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_yBNQd7.pyx..\ . Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_QNaCBv.pyx..\ . Codes for computing mu, or looking it up in mu_dic, a dictionary containing the values of mu for graphs with at most 7 vertices. Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_qygsEw.pyx... Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_yBNQd7.pyx... Compiling /home/sageuser/5/.sage/temp/sage.math.iastate.edu/25083/tmp_QNaCBv.pyx... Codes for computing mu, or looking it up in mu_dic, a dictionary containing the values of mu for graphs with at most 7 vertices. |
def cprism(g):
n=g.order();
h=g.copy();
h.relabel();
hc=h.complement().copy();
newg=h.disjoint_union(hc);
rename={};
for i in range(n):
rename[(0,i)]=-i-1;
rename[(1,i)]=i+1;
newg.relabel(perm=rename);
for i in range(1,n+1):
newg.add_edge((-i,i));
return newg;
def partial_cprism(g,mat=[]):
##g need to be labeled by range(n)
n=g.order();
h=g.copy();
h.relabel();
hc=h.complement().copy();
newg=h.disjoint_union(hc);
rename={};
for i in range(n):
rename[(0,i)]=-i-1;
rename[(1,i)]=i+1;
newg.relabel(perm=rename);
for i in mat:
newg.add_edge((-i-1,i+1));
return newg;
def show_cprism(g,ZFS="ZFS"):
n=g.order();
h=g.copy();
h.relabel();
sage.graphs.graph_plot._circle_embedding(h,h.vertices(),shift=0.1);
pos=h.get_pos();
#for v in h.vertices():
# x,y = pos[v];
#
# # find the angle
# t = arcsin(y)
# if x < 0:
# t = pi/2 - t;
# t += 0.1; # rotate a little
# pos[v] = (cos(t), sin(t));
hc=h.complement().copy();
newg=h.disjoint_union(hc);
matching=[];
rename={};
new_pos={};
for i in range(n):
rename[(0,i)]=-i-1;
rename[(1,i)]=i+1;
newg.relabel(perm=rename);
for i in range(1,n+1):
newg.add_edge((-i,i));
matching.append((-i,i));
new_pos[-i]=(pos[i-1][0]-1.5,pos[i-1][1]);
new_pos[i]=(pos[i-1][0]+1.5,pos[i-1][1]);
newg.set_pos(new_pos);
if ZFS=="ZFS":
ZFS=gZ_leq(cprism(g),i=g.order()-1);
show(newg,figsize=[8,8],vertex_size=200,vertex_colors={"pink":newg.vertices(),"skyblue":ZFS},edge_colors={"gray":matching});
def HadwigerNumber(g):
eta=1;
again=1;
while again:
again=0;
if has_minor(g,graphs.CompleteGraph(eta)):
eta+=1;
again=1;
return eta-1;
def neighborhood_condition(g):
V=g.vertices();
n=len(V);
nbrhood={};
for v in V:
nbrhood[v]=set(g.neighbors(v));
done=False;
for v in V:
for u in V:
if u!=v:
if nbrhood[v].issubset(nbrhood[u]):
#print v,nbrhood[v];
#print u,nbrhood[u];
done=True;
break;
if done:
break;
return done;
def subdivision(g,e):
#input g and e and return the subdivision of g on e
h=g.copy();
i=-1;
while True:
if i in h.vertices():
i-=1;
else:
break;
h.add_vertex(i);
h.delete_edge(e);
h.add_edge((e[0],i));
h.add_edge((e[1],i));
return h;
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