University of Ljubljana

Institute of Mathematics, Physics and MechanicsDepartment of Mathematics

Jadranska 19, 1000 Ljubljana, Slovenia

Preprint series, Vol. 40 (2002), 845

THE LJUBLJANA GRAPH

Marston Conder Aleksander MalnicDragan Marusic Tomaz Pisanski

Primoz Potocnik

ISSN 1318-4865

November 19, 2002

Ljubljana, November 19, 2002

The Ljubljana GraphMarston Conder�- onder�math.au kland.a .nzAleksander Malni� y- Aleksander.Malni �uni-lj.siDragan Maru�si� z- Dragan.Marusi �uni-lj.siToma�z Pisanskix- Tomaz.Pisanski�fmf.uni-lj.siPrimo�z Poto� nik{- Primoz.Poto nik�fmf.uni-lj.siNovember 14, 2002Abstra tA detailed des ription is given of a re ently-dis overed edge- but not vertex-transitivetrivalent graph on 112 verti es, whi h turns out to be the third smallest example of su ha semisymmetri ubi graph. This graph is alled the Ljubljana graph by the authors,although it is believed that its existen e may have been known by R. M. Foster. Withthe help of some advan ed theory of overing graphs, various properties of this graphare analysed, in luding a onne tion with the Heawood graph whi h an be des ribedusing ideals over polynomial rings.x1 Introdu tion In [3℄ I. Z. Bouwer wrote: \R. M. Foster (private ommuni ation)has found an edge- but not vertex-transitive ubi graph (with 112 verti es) whose girth(equal to 10) is not a multiple of 4." Neither its de�nition nor any further informationon this graph was given in that paper. Edge- but not vertex-transitive regular graphs arenow alled semisymmetri . In 2001, during a brief visit by the �rst author to Ljubljana,we onstru ted a ubi semisymmetri graph on 112 verti es, whi h an be des ribed as aregular Z32- over of the Heawood graph. At the �rst author's suggestion, we named thisgraph the Ljubljana graph. A omputer-based sear h showed that the Ljubljana graph isthe only ubi semisymmetri graph on 112 verti es, and hen e the one already known toFoster. Computations also revealed that the Ljubljana graph, whi h throughout this paper�University of Au kland, New Zealand, partially supported by the Marsden Fund ( ontra t no. UOA124)yIMFM, OM, Univerza v Ljubljani, Slovenija, partially supported by Ministrstvo za �solstvo, znanost in�sport, Resear h program 506zIMFM, OM, Univerza v Ljubljani, Slovenija, partially supported by Ministrstvo za �solstvo, znanost in�sport, Resear h program 506xIMFM, OTR, Univerza v Ljubljani, Slovenija, partially supported by Ministrstvo za �solstvo, znanost in�sport, Grants J1-8549-0101,L2-2183-0101{IMFM, OM, Univerza v Ljubljani, Slovenija, partially supported by Ministrstvo za �solstvo, znanost in�sport, Grant Z1-4186-0101 1

is denoted by L, is the third graph in the series of ubi semisymmetri graphs (orderedby number of verti es). The smallest two members of this series are the Gray graph G (on54 verti es), whi h was studied extensively in [10℄ and [11℄, and a biprimitive graph on 110verti es whi h was des ribed in [7℄. The aim of this arti le is to give a pre ise de�nition ofthe Ljubljana graph, to give a omputer-free des ription of its automorphism group, andto present some other interesting graph-theoreti properties of L. The prin ipal means ofexploring the properties of L will be the use of advan ed overing graph te hniques, asdeveloped in [8, 9℄.x2 Arti le layout This arti le is split into several small paragraphs, ea h being a stepin the analysis of the properties of the Ljubljana graph. Sin e the Ljubljana graph is a overing graph of the Heawood graph, x3 to x5 give a short review of the Heawood graphand its automorphism group. In order to make the arti le easier to read, x6 to x13 summarisethe use of overing graph te hniques. This theory is then applied to de�ne the Ljubljanagraph as a Z32- over of the Heawood graph in x14 to x17, and to ompute its automorphismgroup in x18 to x22. Finally, the hamiltoni ity of L is dealt with in x23, two on�gurationsasso iated with the Ljubljana graph are dis ussed in x24, and some additional propertiesof L are gathered together in x25.x3 The Heawood graph The vertex-set of the Heawood graphH onsists of two opiesof the integer ring Z7, so that V (H) = Z7�Z2, and then the edge-set is de�ned by makinga vertex (i; 0) adja ent to a vertex (j; 1) if and only if j � i 2 f1; 2; 4g. We shall abbreviatethe notation by writing i in pla e of (i; 0), and i0 in pla e of (i; 1), for all i 2 Z7.x4 The automorphism group of the Heawood graph Among the automorphismsof the Heawood graph, the following three are easy to �nd:� : (i; j) 7! (i+ 1; j); for i 2 Z7; j 2 Z2,asso iated with the left regular representation of Z7 on itself, and� : (i; j) 7! (2i; j); for i 2 Z7; j 2 Z2,asso iated with the group automorphism of Z7 arising from multipli ation by 2, and� : (i; j) 7! (�i; j + 1); for i 2 Z7; j 2 Z2,whi h inter hanges the two sets of the biparition. These automorphisms may also be easilysee as the following permutations:� = (0; 1; 2; 3; 4; 5; 6)(00 ; 10; 20; 30; 40; 50; 60), 2

� = (0)(1; 2; 4)(3; 6; 5)(00)(10; 20; 40)(30; 60; 50), and� = (0; 00)(1; 60)(2; 50)(3; 40)(4; 30)(5; 20)(6; 10).The group �21 = h�; �i is isomorphi to the semidire t produ t Z7oZ3, and is edge- but notvertex-transitive on H. The group �14 = h�; �i is isomorphi to the dihedral group D7 andis vertex-transitive on H. The group �42 = h�; �; �i is isomorphi to the semidire t produ tZ7oZ6 and is ar -transitive on H. The full automorphism group AutH has order 336 andis isomorphi to PGL(2; 7). It is generated by the involution � and the index 2 subgroupAut0H whi h preserves the two sets of the bipartition. Note that Aut0H is isomorphi toPSL(2; 7) �= GL(3; 2), and that Aut0H = h�; !i, where! = (1; 5)(4; 6)(00 ; 50)(30; 60)(0)(2)(3)(10)(20)(40).Finally, observe that the normalizer NAutH(�) of the group h�i in AutH is the group �42.x5 The Heawood graph as a Cayley graph The subgroup �14 = h�; �i �= D7 ofAutH a ts regularly on the vertex-set of H. This shows that the Heawood graph is aCayley graph of the group h�; �i, with respe t to the generating set f�1; �2; �3g where�1 = �� = (0; 10)(1; 00)(2; 60)(3; 50)(4; 40)(5; 30)(6; 20),�2 = �2� = (0; 20)(1; 10)(2; 00)(3; 60)(4; 50)(5; 40)(6; 30),�3 = �4� = (0; 40)(1; 30)(2; 20)(3; 10)(4; 00)(5; 60)(6; 50).x6 Voltage assignments and derived overing graphs An ordered pair (u; v) ofadja ent verti es of a graph X is alled a dart (or an ar ) of X. If x = (u; v) is a dart ofX then the pair x�1 = (v; u) is the inverse dart of X. Let D be the set of all darts of agraph X and let N be a �nite group. A mapping � : D ! N , satisfying �(x�1) = �(x)�1for every dart x 2 D, is alled a voltage assignment in the voltage group N on the graph X.A voltage assignment � gives rise to the derived overing graph Cov(X; �), with vertex-setV �N and with adja en y relation given by (u; a) � (v; b) if and only if u is adja ent to vin X and a�1b = �(u; v). The graph Cov(X; �) is also alled an N -regular over of X. Theproje tion V �N ! V is learly a graph morphism }� : Cov(X; �)! X, alled the derived overing proje tion. Voltage assignments �; � : D ! N are said to be isomorphi if thereexists an automorphism g 2 AutX and an isomorphism ~g : Cov(X; �) ! Cov(X; �) su hthat }� Æ ~g = g Æ }� . In parti ular, if g an be taken as the identity automorphism of Xthen � and � are said to be equivalent. Note that the derived overing graphs of isomorphi voltage assignments are isomorphi as graphs.x7 Voltage assignments de�ned on the fundamental group Let b0 be a vertex ofa graphX, let �(X; b0) denote its fundamental group, and let � : D ! N be a voltage assign-ment on X. The extension of � to walks indu es a group homomorphism �� : �(X; b0)! N .3

Note that if �� = �� then the voltage assignments �; � are equivalent. Also, for an arbitrarygroup homomorphism � : �(X; b0)! N there exists an assignment � : D ! N with �� = �.When onvenient we an therefore onsider voltage assignments (up to equivalen e) as de-�ned on a fundamental group. By abuse of notation we often write � for ��. Observe thatCov(X; �) is onne ted if and only of �� is surje tive. For the rest of this arti le we willrestri t ourselves to onne ted overing graphs.x8 Abelian voltage assignments If the voltage group N is abelian, then every grouphomomorphism � : �(X; b0) ! N an be fa tored through the abelianisation of �(X; b0),that is, through the �rst homology group H1(H;Z). Further, if p is a prime and N �=Zkp is an elementary abelian group, then the group homomorphism � indu es a Zp-lineartransformation H1(X;Zp) ! N . In this ase, the voltage assignment an be viewed asbeing de�ned on the Zp-ve tor spa e H1(X;Zp). In parti ular, for p = 2 the ve tor spa eH1(X;Zp) oin ides with the y le spa e C(X), and the voltage assignment C(X) ! N isthen uniquely determined by the images of the elements of a basis of C(X).x9 Lifting automorphisms Let }� : Cov(X; �) ! X be the derived overing proje -tion, and let g 2 AutX and ~g 2 AutCov(X; �) be su h that }�~g = g}� . Then we say thatg lifts along }� to ~g, and, that ~g proje ts along }� to g. We also say that }� (respe tively�) is g-admissible in this ase. More generally, if G � AutX then }� (respe tively �) isG-admissible if it is g-admissible for ea h g 2 G. It an be shown that an automorphismg 2 AutX lifts along }� if and only if it maps any losed walk with trivial voltage to a losed walk with trivial voltage. In parti ular, if N �= Zkp is elementary abelian and if thevoltage assignment � : H1(X;Zp)! N is viewed as a Zp-linear mapping, then g lifts along}� if and only if ker � is an invariant subspa e or the indu ed a tion of g on H1(X;Zp).x10 The indu ed linear representation # Let � : H1(X;Zp) ! N be a voltageassignment on X, where N �= Zkp. Suppose that g 2 AutX lifts along the derived overingproje tion }� : Cov(X; �) ! X. By x9 the kernel ker � is an invariant subspa e for theindu ed a tion of g on H1(X;Zp). Sin e N �= H1(X;Zp)= ker �, there exists an indu edautomorphism g# 2 AutN de�ned by g#(�(C)) = �(g(C)), for every C 2 H1(X;Zp).Moreover, it an be shown that if }� is G-admissible, then the mapping #: g 7! g# is agroup homomorphism from G onto a subgroup G# of AutN .x11 Lifting a group Let � : �(X; b) ! N be a G-admissible voltage assignment on X.Then the olle tion of lifts of all elements g 2 G onstitutes a subgroup of AutCov(X; �) alled the lift of G and denoted by ~G. The lift of the trivial group of automorphismsis known as the group of overing transformations or self-equivalen es of }� , and is alsodenoted by CT(}�). Sin e Cov(X; �) is assumed to be onne ted, CT(}�) a ts regularlyon ea h �bre of the form }�1� (v) for v 2 V , or }�1� (x) for x 2 D, and an be identi�ed withthe left regular representation of N on itself. The group CT(}�) is a normal subgroup of~G, with ea h oset being the set of all lifts of one element of G. Thus, ~G=CT(}�) �= G.4

The stru ture of ~G, viewed as an extension of CT(}�) by G, is diÆ ult to analyse. In x12we onsider a very spe ial ase.x12 The stru ture of the lifted group Let � : H1(X;Zp) ! N be a G-admissiblevoltage assignment on a graph X, where N �= Zkp. Let be the G-orbit of a base vertex b0,and let � denote the set of all walks in X with end-verti es in . Suppose further that theelements in G (or just the generators of G) map the trivial voltage walks from � to trivialvoltage walks. Then the lifted group is isomorphi to the semidire t produ t ~G! N o#G,where #: G! AutN is the homomorphism de�ned in x10. This isomorphism is given bythe rule~g 7! (a; g), where a 2 N is de�ned by ~g(b0; 0) = (g(b0); a).Multipli ation in N o# G is given by the rule(a; g)(b; h) = (a+ g#(b); gh) for a; b 2 N; g; h 2 G;and the a tion of N o# G on the vertex set V (X) �N of Cov(X; �) is given by the rule(a; g)(v; b) = (g(v); a+ g#(b)+ g#(�(Wv))� �(g(Wv))) for a; b 2 N; g 2 G; v 2 V (X);where Wv is an arbitrary walk from v to b0. The subgroup N o# f1Gg (isomorphi to N) orresponds to CT(}�) and the subgroup f1Ngo#G (isomorphi to G) orresponds to the omplement whi h maps the verti es above labelled 0 to verti es labelled 0. In parti ular,if G is a subgroup of the stabilizer of b0, then G lifts as a semidire t produ t.x13 De omposing a overing proje tion Let � : H1(X;Zp)! N be a G-admissiblevoltage assignment on X, where N �= Zkp. Let K be a subgroup of N and qK : N ! N=Kthe orresponding quotient proje tion. The following an be shown: If K is invariant forG# then the voltage assignment qK Æ � : H1(X;Zp) ! N=K is G-admissible and thereexists a overing proje tion }K : Cov(X; �)! Cov(X; qK Æ �) su h that }� = }(qKÆ�) Æ }K .Conversely, if � 0 is G-admissible su h that there exists }0 : Cov(X; �) ! Cov(X; � 0) with}� = }�0 Æ}0, then there exists a G#-invariant subgroupK in N su h that � 0 is equivalent toqK Æ�. The voltage assignments qKÆ� and qLÆ� are equivalent if and only ifK = L. Finally,if L = �#(K) for some � 2 AutX whi h lifts along }� , then the voltage assignments qK Æ �and qL Æ � are isomorphi .x14 A Z72- over of the Heawood graph Let H be the Heawood graph as des ribedin x3, and let B be a basis of its y le spa e C(H), onsisting of the following y les:� the Hamilton y le CH : (0; 20; 1; 30; 2; 40; 3; 50; 4; 60; 5; 00; 6; 10; 0), and5

� the seven 6- y les Ci : (i� 2; i0; (i� 1); (i + 1)0; i; (i + 2)0; i� 2); for i 2 Z7.Let N be the additive group of the quotient ring R = Z2[x℄=hx7�1i, viewed as a ve tor spa eof dimension 7 over Z2. By abuse of notation, for an arbitrary polynomial f(x) 2 Z2[x℄, wewill denote the orresponding element f(x) + hx7�1i of the quotient ring by f(x). In viewof x8, let � : C(H)! N be the voltage assignment determined by the rule�(CH) = 0 and �(Ci) = xi for i 2 Z7:We shall denote the derived overing proje tion by ~} : ~H ! H. If one prefers to view thevoltage assignment as being de�ned on darts, then � assigns the trivial element of N to thedarts of the y le CH , and the element xi to the dart (i� 2; (i+ 2)0), for ea h i 2 Z7.

Figure 1: Two drawings of the Heawood graph with the voltage assignment �x15 The lifted subgroup of Aut ~H By x9, an automorphism g of H lifts along ~} ifand only if the kernel of the voltage assignment � is invariant for the indu ed a tion of g onthe y le spa e C(H). In parti ular, sin e the kernel ker � is the 1-dimensional subspa e ofC(H) spanned by the Hamilton y le CH , an automorphism of H lifts along ~} if and onlyif it maps the y le CH onto itself. The maximal subgroup of AutH that lifts along ~} istherefore equal to AutCH \AutH = h�; �i = �14. By x10, the fa t that the automorphisms� and � lift along ~} implies the existen e of linear transformations �#; �# 2 AutN withthe property that �#� = �� and �#� = �� . Expli itly,�#(f(x)) = xf(x) and �#(f(x)) = f(x�1); whenever f(x) 2 N:Observe that a walk in H has trivial voltage if and only if it traverses ea h edge not on-tained in the Hamilton y le CH an even number of times. This property is learly preservedby any element of �14, hen e by x12 the lifted group ~�14 is isomorphi to the semidire tprodu t N o# �14, with multipli ation given by the rule6

(f(x); �i�k)(g(x); �j� l) = (f(x) + xig(x�k); �i+(�1)kj�k+l);whenever f(x); g(x) 2 N; i; j 2 Z7 and k; l 2 Z2. The a tion of No#�14 on the vertex-setV (H)�N of ~H is given by the rule(f(x); �i�k)(v; g(x)) = ((�i�k)(v); f(x) + xig(x�k));whenever f(x); g(x) 2 N; v 2 V (H); i 2 Z7 and k 2 Z2. Sin e �14 a ts regularly onverti es of H, the lifted group ~�14 a ts regularly on the verti es of ~H, and so ~H is a Cay-ley graph for the group ~�14. Re all from x5 that a generating set for the presentationof H as a Cayley graph of �14 may be omprised of �1 = �� , �2 = �2� and �3 = �4� . Itfollows that the generators of ~H as a Cayley graph of ~�14 are the lifts ~�1, ~�2 and ~�3 satisfying~�1(0; 0) = (10; 0), ~�2(0; 0) = (20; 0), and ~�3(0; 0) = (40; x2),that is, the lifts that map the origin (0; 0) 2 V ( ~H) to its neighbours. Expli itly,~�1 = (0; �1) 2 N o# �14, ~�2 = (0; �2) 2 N o# �14, and ~�3 = (x2; �1) 2 N o# �14.x16 De omposing the overing proje tion ~} Let ~} : ~H ! H be the overing pro-je tion as in x14. Let us �nd all h�i-admissible regular overing proje tions } : ~X !H, upto equivalen e, for whi h there exists a regular overing proje tion }0 : H ! ~X satisfying~} = }Æ}0. By x13 we have to �nd all �#-invariant subspa es of the 7-dimensional Z2-ve torspa e N = Z2[x℄=hx7 � 1i. Re all from x15 that �#(f(x)) = xf(x) for all f(x) 2 N . Thisimplies that a subspa e of N is �#-invariant if and only if it is losed under multipli ation byx in the ring R = Z2[x℄=hx7�1i, that is, if and only if it forms an ideal of R. Sin e the idealsof R are generated by the divisors of the polynomial x7�1 = (1+x)(1+x2+x3)(1+x+x3)in Z2[x℄, we obtain exa tly six proper non-trivial ideals of R, namely h1+xi, h1+x2+x3i,h1+x+x3i, h(1+x)(1+x2+x3)i, h(1+x)(1+x+x3)i and h(1+x+x3)(1+x2+x3)i. Inview of x13 there are six pairwise non-equivalent regular h�i-admissible overing proje tions}N=K derived from the voltage assignments qK Æ�, where K is one of the above six ideals ofR and qK : N ! N=K is the natural quotient proje tion. Note that the overing proje tion}N=K is derived from the voltage assignment mapping a ording to the same rule as �,ex ept that its images are now omputed modulo K instead of modulo hx7�1i. Sin e�#(h1 + xi) = h�#(1 + x)i = h1 + x6i = h1 + xi; and then also�#(h1 + x+ x3i) = h�#(1 + x+ x3)i = h1 + x6 + x4i = h1 + x2 + x3i;it follows that the overing proje tions asso iated with h1 + x+ x3i and h1 + x2 + x3i areisomorphi . The same holds for those asso iated with h(1+x)(1+x+x3)i and h(1+x)(1+x2 + x3)i. As a �nal remark, note that N=K �= Zr2, where r 2 f1; 3; 4; 6g is the degree ofthe generating polynomial of the ideal K. 7

x17 The Ljubljana graph as a Z32- over of the Heawood graph With the as-sumptions and the notation of x16, let }L : L ! H be the overing proje tion arising fromthe ideal K = h1 + x2 + x3i. Re all that the voltage group �N = N=K �= Z32 is the additivegroup of the quotient ring Z2[x℄=h1 + x2 + x3i, and the voltage assignment �L = qK Æ � is ongruent to � modulo 1+ x2+ x3 (see Figure 1). The overing graph L is ubi , bipartiteand has order 14 � 23 = 112.x18 The maximal group whi h lifts along }L : L ! H Let M be the maximalsubgroup of AutH whi h lifts along the overing proje tion }L : L ! H as introdu ed inx17. We know that � 2M . By x9 an automorphism g 2 AutH belongs toM if and only if itpreserves the kernel of �L : C(H)! Z32. Clearly, the kernel ker �L onsists of all elements ofC(H) with �-voltages in K, and has dimension 5. Now set W = C0 +C2 +C3 and observethat �(�i(W )) 2 K for ea h i 2 Z7. Sin e fCH ; �(W ); �2(W ); �3(W ); �3(W ); �4(W )g islinearly independent it follows that ker �L is spanned by CH and �i(W ), for i 2 Z7. Astraightforward al ulation gives�(CH) =Xi2Z7�i(W ) and �(�i(W )) = �4i(�(W )) = CH + �4i(W );whi h implies that � 2M . On the other hand, �(Ci) = C�i for i 2 Z7, and so �L(�(W )) =�L(C0 + C1 + C3) = 1 + x+ x3 6= 0. Similarly, �L(!(CH)) = x2 6= 0. Hen e �; ! 62 M . Wenow prove that M = h�; �i = �21. As already shown, �21 �M , and hen e jM j = 3 � 7 � 2r,where r 2 f0; 1; 2; 3; 4g. If r = 4 then M = AutH, ontradi ting the fa t that � 62 M .If r = 3 then M is a normal subgroup of index 2 in AutH and M \ Aut0H is normal inAut0H �= PSL(2; 7). Sin e this interse tion is non-trivial and sin e PSL(2; 7) is simple,we �nd M = Aut0H, ontradi ting the fa t that ! 62 M . Let k be the number of Sylow7-subgroups of M . By Sylow's theorem, k � 1 mod 7, and k divides 3 �2r . If k > 1, then wehave r � 3, a ontradi tion. This shows that M is ontained in the normalizer NAutH(�)of h�i in AutH. Next re all from x4 that NAutH(�) = h�; �; �i = �42. Sin e � 62 M , thegroup M is a proper subgroup of �42, and so M = �21, as required.x19 The girth of L We show that the girth of L is 10, and that there are exa tly3 � 7 � 23 = 168 y les of length 10. To this end suppose that C is a y le in L of lengthn � 10, and let C 0 = }L(C) be its proje tion. Sin e the girth of the Heawood graph His 6, C 0 is also a y le of length 6, 8 or 10. As the ar s of H with trivial voltages indu ea Hamilton y le (of length 14), C 0 ontains at least one ar with non-trivial voltage.Now besides the zero polynomial 0 and the y lotomi polynomial 1 + x + : : : + x6, theideal h1 + x2 + x3i of Z2[x℄ ontains seven elements of the form xi + xi+2 + xi+3 andseven elements of the form xi + xi+3 + xi+4 + xi+5, where i 2 Z7. Sin e the sum ofvoltages of edges of C 0 is the trivial element of the ring Z2[x℄=h1 + x2 + x3i, it followsthat there is some i 2 Z7 su h that the set of the non-trivial voltages of the edges of C 0is either fxi; xi+2; xi+3g or fxi; xi+3; xi+4; xi+5g. A loser inspe tion of the y le stru tureof H then reveals that the length of C 0 is 10, and moreover, that it belongs to one of the8

three orbits of the automorphism � ontaining the y les A1 = (0; 40; 3; 50; 1; 20; 5; 00; 6; 10; 0),A2 = (0; 20; 5; 60; 4; 50; 1; 30; 2; 40; 0) and B = (1; 50; 4; 60; 2; 40; 3; 00; 5; 20; 1). Sin e ea h of these y les lifts to 23 disjoint 10- y les in L and ea h of these three �-orbits ontains seven y les,the total number of 10- y les in L is 3 � 7 � 23 = 168.Figure 2: The three types of 10- y les of H and Lx20 Cy les of length 12 in L Let C be a y le in L of length 12, and let C 0 = }L(C)be its proje tion. As the girth of H is 6, the graph C 0 an be either a 6- y le or a 12- y lewith trivial voltage. Observe that there are four �-orbits of 6- y les inH, with the followingrespe tive representatives:D1 = (0; 20; 1; 30; 2; 40; 0),D2 = (0; 40; 2; 60; 4; 10; 0),D3 = (0; 10; 4; 50; 1; 20; 0),D4 = (0; 20; 5; 60; 4; 10; 0).Observe further that ea h of these y les lifts to exa tly four 12- y les in L, and that if C 0is a 12- y le then the voltage of C 0 must be trivial. By a similar argument as used in x19,we an show that all 12- y les with trivial voltage in H belong to the same �-orbit, withrepresentativeE = (0; 40; 2; 30; 1; 20; 5; 00; 3; 50; 4; 10; 0).A lift of a y le in the �-orbits of Di (or E) will be referred to as a y le of type Di (or

Figure 3: The �ve types of 12- y les of Lof type E, respe tively). Sin e ea h 6- y le of type Di lifts to four 12- y les while ea h9

12- y le of type E lifts to eight 12- y les, it follows that there are (4 � 4+8) � 7 = 168 y lesof length 12 in the Ljubljana graph L.x21 Blo ks of imprimitivity of L For a vertex u of H, let the �bre of u be the setof eight verti es of L whi h proje t by }L to u. In this Paragraph we prove that �bresare blo ks of imprimitivity of the a tion of AutL on the verti es of L. First observe thattwo antipodal verti es of a y le of type E lie on a ommon 10 y le of L. (It suÆ esto he k this for lifts of E; for example the lifts of verti es 0 and 5 lie on a lift of the 10 y le �(B).) Next observe that the 12- y les of types D1, D2 and D3 do not have thisproperty. This implies that there are no automorphisms of L mapping a 12- y le of typeE to a 12- y le of type D1, D2 or D3. Let us now de�ne an equivalen e relation R on thevertex-set of L by the rule: uR v if and only if u = v or u and v are antipodal verti es of a12- y le of type Di, for i 2 f1; 2; 3g. The above-mentioned fa t about 10- y les shows thatR is an (AutL)- ongruen e, whi h implies that the equivalen e lasses of R are blo ks ofimprimitivity for AutL. As it is easy to see that the equivalen e lasses of R are exa tlythe �bres of the overing proje tion }L : L ! H, our laim follows.x22 The stru ture and the a tion of AutL By x21, the �bres of }L : L ! H areblo ks of imprimitivity of AutL, and so AutL proje ts along }L. By x18, the maximalgroup that lifts along }L is the meta y li group �21 = h�; �i. Hen e AutL = ~�21 is anextension by �21 of the additive group �N of the quotient ring Z2[x℄=h1 + x2 + x3i. Asj~�21j = 21 and �N �= Z32, the group AutL has order 23 � 3 � 7 = 168. Moreover, sin e �21 a tsedge- but not vertex-transitively on H, the Ljubljana graph is semisymmetri . Next re allthat there is a natural embedding � : �N ! AutL, given by �(a)(v; b) = (v; a + b), whi hmaps �N isomorphi ally onto the group of overing transformations. Clearly the extension�N ! AutL ! �21 splits, by the Zassenhaus lemma [12℄. For the the sake of ompletenesswe give a self ontained proof that AutL �= �N o#�21, and give its expli it a tion on vertexset V (H)� �N of L. First, by x10 there exists a homomorphism #: �21 ! Aut �N satisfyingg#�L = �Lg for every g 2 �21. An expli it al ulation involving the generators Ci de�nedin x14 for C(H) shows that�#(f(x)) = xf(x) and �#(f(x)) = x6f(x2) whenever f(x) 2 �N:Let ~� and ~� be the respe tive lifts of � and � mapping the vertex (0; 0) 2 V (H) � �N toverti es (1; 0) and (0; x6), respe tively. In view of x12, the a tions of the automorphisms~�; ~� 2 ~�21 on V (H)� �N are given by the rules~�(v; f(x)) = (�(v); xf(x)) and ~�(v; f(x)) = (�(v); x6 + x6f(x2) + �L(�(Wv)));where Wv is a trivial voltage walk in H from v to 0. To al ulate the values �L(�(Wv)) we an hoose the walks Wv to be ontained in the Hamilton y le CH . Sin e the automor-phism � maps CH to the y le (0; 40; 2; 60; 4; 10; 6; 30; 1; 50; 3; 00; 5; 20; 0), the values �L(�(Wv))10

are partials sums of the sequen e x2; x4; x6; x8; x10; x12; x14(= 1). More pre isely,�L(�(Ws)) = �L(�(W(s+1)0)) = x2 + x4 + : : :+ x2s = x6(1 + x2s); for s 2 Z7:Sin e ����1 = �2 it follows that there exists an element a 2 �N su h that ~�~�~��1 = �(a)~�2.The element a an be omputed using the equality ~�~� = �(a)~�2~� at the vertex (0; 0):First ~�(~�(0; 0)) = ~�(1; 0) = (2; x6 + x6 � 0 + x6(1 + x2)) = (2; x);and se ond �(a)(~�2(~�(0; 0))) = �(a)(~�2(0; x6)) = �(a)(2; x):This shows that a = 0, and so ~� normalizes ~�. Consequently, the group h~�; ~�i is isomorphi to �21 and is a omplement of �( �N ) � ~�21. The automorphism group AutL is thereforean internal semidire t produ t �( �N) o h~�; ~�i; isomorphi also to the external semidire tprodu t �N o# h�; �i with multipli ation given by the rule(f(x); �i�k)(g(x); �j�l) = (f(x) + x�2k+i+1g(x2k); �i+2kj�k+l):x23 Hamiltoni ity of L and its LCF ode The LCF ode [6℄ of a hamiltonian ubi graph relative to one of its Hamilton y les (v0; v1; v2; : : : ; vn�1; v0) is a list LCF[a0; a1; : : : ; an�1℄of elements of Zn n f0; 1; n� 1g su h that vi is adja ent to vi+ai for every i 2 Zn. In addi-tion, if there exists a proper divisor r of n su h that ai = ai+kr for all i 2 f0; 1; : : : ; r�1gand k 2 f1; 2; : : : ; nr �1g, then the notation an be simpli�ed to LCF[a0; a1; : : : ; ar�1℄nr . Inthis ase we say the LCF ode is periodi , and all the integers r and nr the period and theexponent of the LCF ode, respe tively. When sear hing for a Hamilton y le of L witha periodi LCF ode, the following observations are immediate. First, the non-identityelements of AutL have orders 2, 3, 6 and 7. In parti ular, AutL ontains no semiregularelements of order 3 or 6 (semiregular meaning that all y les of the permutation have thesame length), and therefore there is no LCF ode of L with exponent 3 or 6. On the otherhand, AutL does ontain semiregular elements of orders 2 and 7. All elements of order 7are onjugate however, and as is easily seen from Figure 6, the quotient graph of L relativeto an element of order 7 is not hamiltonian. Consequently, L has no LCF ode of exponent7. The only remaining possibilities for the LCF odes of L are therefore an aperiodi LCF ode, orresponding to a Hamilton y le whi h is �xed by no automorphism of L, and anLCF ode of exponent 2, orresponding to a Hamilton y le �xed by a semiregular invo-lution, that is, an element in the group of overing transformations Z32. It transpires thatboth possibilities do o ur. The following is an aperiodi LCF ode for L, whi h gives riseto the �rst drawing of L given in Figure 4:LCF[11; 55;�23; 31; 11;�9; 55; 17; 39;�23; 31;�11; 9;�31;�23;�11;�53; 31;�47; 35;11;�9; 55; 13;�17;�45; 17; 47;�29;�39;�53;�11; 13; 39;�31; 49;�13; 21;�55; 49;13;�31; 25;�17; 35;�13; 11;�39;�31;�51; 21; 29; 49;�13;�35; 51;�55;�11;�21; 9;21;�55; 11; 23; 29; 35; 43;�25;�9; 21; 35;�21;�39;�11;�47; 53; 23;�55; 9;�35;11

Figure 4: Three drawings of the Ljubljana graph: the �rst one exhibiting an asymmetri Hamilton y le, the se ond exhibiting a Hamilton y le with entral symmetry, and thethird exhibiting the existen e of a semiregular automorphism of order 7�29;�21; 15; 47;�49; 19;�23;�9;�49; 53;�21; 23; 45;�29; 31; 55; 11;�15; 23;�23;�35;�49; 39; 23;�19;�35;�51;�11; 9;�43; 51; 29℄On the other hand, the LCF ode given below has exponent 2, and therefore gives rise toa entral symmetry of L as shown in the se ond drawing in Figure 4:LCF[47;�23;�31; 39; 25;�21;�31;�41; 25; 15; 29;�41;�19; 15;�49; 33; 39;�35;�21;17;�33; 49; 41; 31;�15;�29; 41; 31;�15;�25; 21; 31;�51;�25; 23; 9;�17; 51; 35;�29;21;�51;�39; 33;�9;�51; 51;�47;�33; 19; 51;�21; 29; 21;�31;�39℄2x24 A dual pair of Ljubljana on�gurations A (v3)- on�guration is an in iden estru ture having v points and v lines with the following properties: ea h line ontainsexa tly 3 points, ea h point belongs to exa tly 3 lines, and any two lines interse t in atmost one point. Su h a on�guration determines a ubi bipartite graph with a bla k andwhite vertex olouring, where bla k verti es orrespond to points and white verti es to linesof the on�guration. Further, two verti es are adja ent if and only if they orrespond toan in ident point-line pair. Su h a graph is alled the Levi graph of the on�guration (see12

[4, 5℄). Conversely, ea h ubi bipartite graphon 2v verti es of girth at least 6 is the Levigraph of a dual pair of (v3)- on�gurations, and these two on�gurations are isomorphi if and only if there is an automorphism of their Levi graph whi h inter hanges the twobipartite sets. Sin e L is bipartite of girth 10 and semisymmetri it gives rise to a dual pairof non-isomorphi quadrangle-free on�gurations, shown in Figure 5. Observe that the two

Figure 5: The Ljubljana and the Dual Ljubljana (563) on�gurationsdrawings exhibit a rotational symmetry of order 7. Ea h was produ ed by applying theoryof poly y li on�gurations as outlined in [4℄. The reason that su h theory an be used inthis ontext resides in the fa t that L is a Z7- over of a bipartite base graph (see Figure 6).Indeed as we have seen, L has a semiregular automorphism of order 7, and the quotientgraph with respe t to this is bipartite.6

2

2

6

6

2

2

6

2

2

6

2

2

2

6

6

Figure 6: The Ljubljana graph L is a Z7 over over this graphx25 Some further properties of the Ljubljana graph Here we gather some fur-ther interesting properties of L. First, if one wants to he k that the Ljubljana graph isnot vertex-transitive it suÆ es to show that there exist two verti es with di�erent distan esequen es (the sequen es whi h give the numbers of verti es at su essive distan es from agiven vertex). A dire t omputation shows that the verti es in one set of the bipartitionhave have distan e sequen e (1; 3; 6; 12; 24; 34; 24; 7; 1), while the verti es in the other bi-partition set have distan e sequen e (1; 3; 6; 12; 24; 34; 25; 7).13

Next, sin e the automorphism group of L has order 23:3:7 = 168, and L is edge-transitivewith 168 edges, it follows that the line graph L(L) is a graphi al regular representation(GRR) for the group AutL. More pre isely, L(L) is a Cayley graph for the group of order168 with presentation hx; y j x3 = y3 = xyxy�1xyx�1y�1x�1y�1 = 1 i.Finally, for a graph X let X(2) denote the graph on the same vertex-set as X, and withtwo verti es adja ent in X(2) if and only if they are at distan e 2 in X. If X is onne tedand bipartite the graph X(2) has exa tly two onne ted omponents, X(2)1 and X(2)2 , orre-sponding to the two sets of the bipartition. In the ase where X is the Levi graph of a pairof dual on�gurations C and Cd, the graphs M(C) = X(2)1 and M(Cd) = X(2)2 are alled theMenger graphs of C and Cd, respe tively.The Menger graphs asso iated with the pair of Ljubljana on�gurations are non-isomorphi 12 -ar -transitive Cayley graphs for the group ~�7 �= Z32oZ7 of order 56 (the lift of the y li group h�i as des ribed in x22), and with automorphism group isomorphi to AutL. Thesetwo graphs may be represented as Cayley graphs Cay(~�7; ft; t�1; tx; (t�1)x; tx2 ; (t�1)x2g)and Cay(~�7; ft; t�1; ty; (t�1)y; ty2 ; (t�1)y2g); where x and y are the two elements of order3 introdu ed above as generators for AutL, while t = xy�1. In ea h of these two Mengergraphs, every vertex belongs to three triangles, whi h are y li ally permuted by the orre-sponding vertex stabilizer in the automorphism group. Also sin e tx = ty = y�1x, the twographs ontain a ommon 4-valent GRR of ~�7, namely the graph Cay(~�7; ft; t�1; tx; (t�1)xg).x26 A knowledgements The authors wish to thank Marko Boben for onstru tingthe drawings of the on�gurations in Figure 5, and to a knowledge the use of the Magmasystem [2℄ in determining some of the properties of the Ljubljana graph L de ribed in thispaper.Referen es[1℄ N. L. Biggs, A. T. White, Permutation Groups and Combinatorial Stru tures, LMS Le ture NoteSeries, vol.33, Cambridge University Press, 1979.[2℄ W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symboli Comput. 24 (1997), 235{265.[3℄ I. Z. Bouwer, On edge but not vertex transitive regular graphs, J. Combin. Theory (B) 12 (1972),32{40.[4℄ M. Boben, T. Pisanski, Poly y li on�gurations, Europ. J. Combin., to appear.[5℄ H. S. M. Coxeter, Self-Dual on�gurations and regular graphs, Bull. Amer. Math. So . 56 (1950),413{455.[6℄ R. Fru ht, A anoni al representation of trivalent Hamiltonian graphs, J. Graph Theory 1 (1976),45{60. 14

[7℄ A. A. Ivanov, and M. E. Io�nova, Biprimitive ubi graphs, Investigations in the Algebrai Theory ofCombinatorial Obje ts, 123{134, Vsesoyuz. Nau hno-Issled. Inst. Sistem. Issled., Mos ow, (1985).(2002), 137{152.[8℄ A. Malni� , R. Nedela, and M. �Skoviera, Lifting graph automorphims by voltage assignments, Europ.J. Combin. 21 (2000), 927{947.[9℄ A. Malni� , D. Maru�si� , and P. Poto� nik, Elementary abelian overs of graphs, preprint 2001, submitted.[10℄ D. Maru�si� , T. Pisanski, The Gray graph revisited, J. Graph Theory 35 (2000), 1{7.[11℄ D. Maru�si� , T. Pisanski, S. Wilson, The genus of the Gray graph is seven, submitted.[12℄ J. J. Rotman, An Introdu tion to the Theory of Groups (4th ed.), Springer-Verlag, New York, 1995.

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