Odd automorphisms in vertex-transitive graphs

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1 Also avalable at ISSN (prnted edn.), ISSN (electronc edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) Odd automorphsms n vertex-transtve graphs Ademr Hujdurovć, Klavdja Kutnar Unversty of Prmorska, UP IAM, Muzejsk trg 2, 6000 Koper, Slovena Unversty of Prmorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovena Dragan Marušč Unversty of Prmorska, UP IAM, Muzejsk trg 2, 6000 Koper, Slovena Unversty of Prmorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovena IMFM, Jadranska 19, 1000 Ljubljana, Slovena Receved 24 February 2016, accepted 10 July 2016, publshed onlne 25 July 2016 Abstract An automorphsm of a graph s sad to be even/odd f t acts on the set of vertces as an even/odd permutaton. In ths artcle we pose the problem of determnng whch vertex-transtve graphs admt odd automorphsms. Partal results for certan classes of vertex-transtve graphs, n partcular for Cayley graphs, are gven. As a consequence, a characterzaton of arc-transtve crculants wthout odd automorphsms s obtaned. Keywords: Graph, vertex-transtve, automorphsm group, even permutaton, odd permutaton. Math. Subj. Class.: 20B25, 05C25 1 Introducton Apart from beng a rch source of nterestng mathematcal objects n ther own rght, vertex-transtve graphs provde a perfect platform for nvestgatng structural propertes of transtve permutaton groups from a purely combnatoral vewpont. The recent outburst Ths work s supported n part by the Slovenan Research Agency (research program P and research projects N1-0032, N1-0038, and J1-7051). Ths work s supported n part by the Slovenan Research Agency (research program P and research projects N1-0032, N1-0038, J1-6720, J1-6743, and J1-7051), n part by WoodWsdom-Net+, W 3 B, and n part by NSFC project Ths work s supported n part by the Slovenan Research Agency (I0-0035, research program P and research projects N1-0032, N1-0038, J1-5433, J1-6720, and J1-7051), and n part by H2020 Teamng InnoRenew CoE. E-mal address: ademr.hujdurovc@upr.s (Ademr Hujdurovć), klavdja.kutnar@upr.s (Klavdja Kutnar)dragan.marusc@upr.s (Dragan Marušč) cb Ths work s lcensed under

2 428 Ars Math. Contemp. 10 (2016) of research papers on ths topc should therefore come as no surprse. Most of these papers have arsen as drect attempts - by developng consstent theores and strateges to solve open problems n vertex-transtve graphs; the hamltoncty problem [16], for example, beng perhaps the most popular among them. In ths context knowng the full (or as near as possble) automorphsm group of a vertextranstve graph s mportant because t provdes the most complete descrpton of ts structure. Whle some automorphsms are obvous, often part of the defnng propertes, there are others, not so obvous and hence more dffcult to fnd. Consder for example bcrculants, more precsely, n-bcrculants, that s, graphs admttng an automorphsm ρ wth two orbts of sze n 2 and no other orbts. There are three essentally dfferent possbltes for such a graph to be vertex-transtve dependng on whether ts automorphsm group contans a swap and/or a mxer, where a swap s an automorphsm nterchangng the two orbts of ρ, and a mxer s an automorphsm whch nether fxes nor nterchanges the two orbts of ρ. For example, the Petersen graph has swaps and mxers, prsms (except for the cube) have only swaps, whle the dodecahedron has only mxers. Clearly, swaps are the obvous automorphsms and mxers are not so obvous ones (see Fgure 1). Fgure 1: The Petersen graph, the 5-prsm and the dodecahedron the frst two admt a swap, whle the thrd one does not. In ths paper we propose to approach the sometmes elusve separaton lne between the obvous and not so obvous automorphsms va the even/odd permutatons dchotomy. Let us call an automorphsm of a graph even/odd f t acts on the vertex set as an even/odd permutaton. Further, a graph s sad to be even-closed f all of ts automorphsms are even. The Petersen graph and odd prsms have odd automorphsms, the swaps beng such automorphsms. On the other hand, the dodecahedron has only even automorphsms [14]. Furthermore, consder the two cubc 2k-bcrculants, k > 1, shown n Fgure 2 for k = 4. Both have swaps whch are even automorphsms. More precsely, all of the automorphsms of the 2k-prsm on the left-hand sde are even. As for the graph on the rght-hand sde the Cayley graph Cay(Z 4k, {±1, 2k}) on the cyclc group Z 4k = 1 any generator of the left regular representaton of Z 4k s an odd automorphsm (note that the bcrculant structure of ths graph arses from the acton of the square of any generator of the left regular representaton of Z 4k ). Ths brngs us to the followng natural queston: Gven a transtve group of even automorphsms H of a graph X, s there a group G Aut(X) contanng odd automorphsms

3 A. Hujdurovć et al.: Odd automorphsms n vertex-transtve graphs 429 Fgure 2: Two examples of cubc 2k-bcrculants for k = 4, one wth and one wthout odd automorphsms. of X and H as a subgroup? In partcular, we would lke to focus on the followng problem. Problem 1.1. Whch vertex-transtve graphs admt odd automorphsms? Of course, n some cases, the answer to the above problem wll be purely arthmetc. Such s for example the case wth cycles. Clearly, all cycles of even length admt odd automorphsms, whle cycles of odd length 2k + 1 admt odd automorphsms f and only f k s odd. The answer for some of the well studed classes of graphs, however, suggest that the above even/odd queston goes beyond smple arthmetc condtons and s lkely to uncover certan more complex structural propertes. For example, whle the general dstngushng feature for cubc symmetrc graphs (wth respect to the above queston) s ther order 2n, n even/odd, there are exceptons on both sdes. Namely, there exst cubc symmetrc graphs wthout odd automorphsms for n odd, and wth odd automorphsms for n even, see [14]. In ths paper a specal emphass s gven to certan classes of Cayley graphs (see Secton 3), such as crculants for example. Theorem 3.15 gves a necessary and suffcent condton for a normal crculant to be even-closed. Ths result combned together wth certan other results of ths secton then leads to a characterzaton of even-closed arc-transtve crculants, see Theorem In Secton 4 the even/odd queston s dscussed n the more general context of vertex-transtve graphs. 2 Prelmnares Here we brng together defntons, notaton and some results that wll be needed n the remanng sectons. For a fnte smple graph X let V (X), E(X), A(X) and Aut(X) be ts vertex set, ts edge set, ts arc set and ts automorphsm group, respectvely. A graph s sad to be vertex-transtve, edge-transtve and/or arc-transtve (also symmetrc) f ts automorphsm group acts transtvely on the set of vertces, the set of edges, and/or the set of arcs of the graph, respectvely. A non-dentty automorphsm s semregular, n partcular (m, n)- semregular f t has m cycles of equal length n n ts cycle decomposton, n other words m orbts of equal length n. An n-crculant (crculant, n short) s a graph admttng a (1, n)-semregular automorphsm, and an n-bcrculant (bcrculant, n short) s a graph admttng a (2, n)-semregular automorphsm.

4 430 Ars Math. Contemp. 10 (2016) Gven a group G and a symmetrc subset S = S 1 of G \ {1}, the Cayley graph X = Cay(G, S) has vertex set G and edges of the form {g, gs} for all g G and s S. Every Cayley graph s vertex-transtve but there exst vertex-transtve graphs that are not Cayley, the Petersen graph beng the smallest such graph. Cayley graphs are characterzed n the followng way. A graph s a Cayley graph of a group G f and only f ts automorphsm group contans a regular subgroup G L, referred to as the left regular representaton of G, somorphc to G, see [23]. Usng the termnology and notaton of Cayley graphs, note that an n-crculant s a Cayley graph Cay(G, S) on a cyclc group G of order n relatve to some symmetrc subset S of G \ {d}, usually denoted by Crc(n, S). The frst of the two group-theoretc observatons below reduces the queston of exstence of odd automorphsms to Sylow 2-subgroups of the automorphsm group. Proposton 2.1. A permutaton group G contans an odd permutaton f and only f ts Sylow 2-subgroups contan an odd permutaton. Proof. Snce any odd permutaton α s of even order, we can conclude that α k, where k s the largest odd number dvdng the order of α, s a non-trval odd permutaton belongng to a Sylow 2-subgroup of G. Proposton 2.2. A permutaton group G actng semregularly wth an odd number of orbts admts odd permutatons f and only f ts Sylow 2-subgroups are cyclc and non-trval. Proof. Note that any Sylow 2-subgroup of G must also have an odd number of orbts. Thus f a Sylow 2-subgroup s cyclc and non-trval, the correspondng generators are odd permutatons. On the other hand, f a Sylow 2-subgroup J s not cyclc (or s trval) then the semregularty of G mples that all of the elements of J must be even permutatons. By Proposton 2.1 G tself conssts solely of even permutatons. As a consequence of Proposton 2.2, for some classes of graphs the exstence of odd automorphsms s easy to establsh. For nstance, n Cayley graphs the correspondng regular subgroup contans odd automorphsms f and only f ts Sylow 2-subgroup s cyclc and non-trval. When a Sylow 2-subgroup s not cyclc, however, the search for odd automorphsms has to be done outsde ths regular subgroup, rasng the complexty of the problem. 3 Cayley graphs In ths secton we gve some general results about the exstence of odd automorphsms n Cayley graphs and dscuss the problem n detal for crculants. The frst proposton, a corollary of Proposton 2.2, gathers straghtforward facts about the exstence of odd automorphsms n Cayley graphs. (A graph s sad to be a graphcal regular representaton, or a GRR, for a group G f ts automorphsm group s somorphc to G and acts regularly on the vertex set of the graph.) Proposton 3.1. A Cayley graph on a group G admts an odd automorphsm n G L f and only f G has cyclc Sylow 2-subgroups. In partcular, a Cayley graph of order 2 (mod 4) admts odd automorphsms, a GRR admts an odd automorphsm f and only f the Sylow 2-subgroups of G are cyclc.

5 A. Hujdurovć et al.: Odd automorphsms n vertex-transtve graphs 431 By Proposton 3.1, Cayley graphs of order twce an odd number admt odd automorphsms (they exst n a regular subgroup of the automorphsm group). As for Cayley graphs whose order s odd or dvsble by 4 the answer s not so smple. The next proposton answers the queston of exstence of odd automorphsms n partcular subgroups of automorphsms of Cayley graphs on abelan groups. Proposton 3.2. Let X = Cay(G, S) be a Cayley graph on an abelan group G and let τ Aut(G) be such that τ() =. Then G L, τ Aut(X), and there exsts an odd automorphsm n G L, τ f and only f one of the followng holds: () G 3 (mod 4) (n whch case τ s an odd automorphsm), () G 2 (mod 4), () G 0 (mod 4) and a Sylow 2-subgroup of G s cyclc. Proof. Frst recall that the mappng τ : G G defned by τ() = s an automorphsm of the group G f and only f G s abelan. Moreover, snce S = S t s easy to see that τ Aut(X). Clearly, when G 1 (mod 4) there are no odd automorphsms n G L, τ. Suppose now that G 1 (mod 4). If G 3 (mod 4) then the nvoluton τ has 2k + 1 cycles of length 2 and one fxed vertex n ts cyclc decomposton, and so t s an odd automorphsm. If G 2 (mod 4) then there exst odd automorphsms n G L G L, τ by Proposton 3.1. We are therefore left wth the case G 0 (mod 4). Hence suppose that G s of such order. If a Sylow 2-subgroup J of G L s cyclc then a generator of J s a product of an odd number G / J of cycles of length J, and s thus an odd automorphsm. On the other hand, f J s not cyclc then every element of J has an even number of cycles n ts cyclc decomposton. As for τ, an element of G s fxed by τ f and only f t s an nvoluton. In other words, t fxes the largest elementary abelan 2-group T nsde the Sylow 2-subgroup J, say of order 2 k. Consequently, the number of transpostons n the cyclc decomposton of τ equals G /2 2 k, whch s an even number f and only f k 1. Consequently, τ s an odd automorphsm f and only f T = Z 2 and hence J s cyclc. Corollary 3.3. Let X = Crc(n, S), where S s a symmetrc subset of Z n, and ether n s even or n 3 (mod 4). Then X admts odd automorphsms. When n 1 (mod 4) the stuaton s more complex. For example, cycles C 4k+1 = Crc(4k + 1, {±1}) admt only even automorphsms. On the other hand, the crculant Crc(13, {±1, ±5}) s an example of a (4k + 1)-crculant admttng odd automorphsms. Namely, one can easly check that the permutaton (0)(1, 5, 12, 8)(2, 10, 11, 3)(4, 7, 9, 6) arsng from the acton of 5 Z 13 s one of ts odd automorphsms. (For a postve nteger n we use Z n to denote the multplcatve group of unts of Z n.) We therefore propose the followng problem. Problem 3.4. Classfy even-closed crculants of order n 1 (mod 4). A partal answer to ths problem s gven at the end of ths secton, see Corollary 3.11 and Theorem We start wth the class of connected arc-transtve crculants. The classfcaton of such crculants, obtaned ndependently by Kovács [10] and L [15], s essental to ths end. In order to state the classfcaton let us recall the concept of normal Cayley graphs and certan graph products.

6 432 Ars Math. Contemp. 10 (2016) Let X and Y be graphs. The wreath (lexcographc) product X[Y ] of X by Y s the graph wth vertex set V (X) V (Y ) and edge set {{(x 1, y 1 ), (x 2, y 2 )}: {x 1, x 2 } E(X), or x 1 = x 2 and {y 1, y 2 } E(Y )}. The deleted wreath (deleted lexcographc) product X d Y of X by Y s the graph wth vertex set V (X) V (Y ) and edge set {{(x 1, y 1 ), (x 2, y 2 )}: {x 1, x 2 } E(X) and y 1 y 2, or x 1 = x 2 and {y 1, y 2 } E(Y )}. If Y = K b = bk 1 then the deleted lexcographc product X d Y s denoted by X[K b ] bx. Let X = Cay(G, S) be a Cayley graph on a group G. Denote by Aut(G, S) the set of all automorphsms of G whch fx S setwse, that s, Aut(G, S) = {σ Aut(G) S σ = S}. It s easy to check that Aut(G, S) s a subgroup of Aut(X) and that t s contaned n the stablzer of the dentty element d G. Followng Xu [25], X = Cay(G, S) s called a normal Cayley graph f G L s normal n Aut(X), that s, f Aut(G, S) concdes wth the vertex stablzer d G. Moreover, f X s a normal Cayley graph, then Aut(X) = G L Aut(G, S) (see [9]). Proposton 3.5. [10, 15] Let X be a connected arc-transtve crculant of order n. Then one of the followng holds: () X = K n ; () X = Y [K d ], where n = md, m, d > 1 and Y s a connected arc-transtve crculant of order m; () X = Y [K d ] dy, where n = md, d > 3, gcd(d, m) = 1 and Y s a connected arc-transtve crculant of order m; (v) X s a normal crculant. The proof of the next proposton s straghtforward. Proposton 3.6. Complete graphs K n and ther complements K n, n 2, admt odd automorphsms. Propostons 3.7, 3.8, 3.9, and 3.10 deal wth the exstence of odd automorphsms n the framework of (deleted) lexcographc products of graphs. Proposton 3.7. Let Z be a graph admttng an odd automorphsm. Then a lexcographc product Y [Z] of the graph Z by a graph Y admts odd automorphsms. In partcular, Y [K d ], d > 1, admts odd automorphsms. Proof. An odd automorphsm s constructed by takng a map that acts trvally on all blocks (that s, copes of the graph Z) but one, where t acts as an odd automorphsm of the graph Z. By Proposton 3.6, K d admts an odd automorphsm, so such a map exsts when Z = K d. Proposton 3.8. Let X be the deleted lexcographc product X = Y d Z of a graph Y by a graph Z, where Z has odd automorphsms and Y s of odd order. Then X admts odd automorphsms. Proof. An odd automorphsm s constructed by takng a map that acts as the same odd automorphsm on each of the odd number of copes of the graph Z.

7 A. Hujdurovć et al.: Odd automorphsms n vertex-transtve graphs 433 Proposton 3.9. Let X be the deleted lexcographc product X = Y d Z of a graph Y by a graph Z, where Z s of odd order and Y has odd automorphsms. Then X admts odd automorphsms. Proof. Let α be an odd automorphsm of Y. Let α : V (X) V (X) be defned wth α((y, z)) = (α (y), z). It s easy to see that α Aut(X), and the fact that V (Z) s odd mples that α s an odd automorphsm of X. Propostons 3.8 and 3.9 combned together mply exstence of odd automorphsms n arc-transtve crculants belongng to the famly gven n Proposton 3.5(). Proposton Let X be an arc-transtve crculant somorphc to the deleted lexcographc product Y [dk 1 ] dy, where Y s an arc-transtve crculant of order coprme wth d > 1. Then X has an odd automorphsm. Proof. Suppose frst that Y s of odd order. Then, snce, by Proposton 3.6, dk 1 admts odd automorphsms, the exstence of odd automorphsms n Aut(X) follows from Proposton 3.8. Suppose now that Y s of even order. Then any generator of a regular cyclc subgroup of Aut(Y ) s an odd automorphsm. Snce n ths case d s odd the exstence of odd automorphsms n Aut(X) follows from Proposton 3.9. Corollary 3.3 and Propostons 3.6, 3.7 and 3.10 combned together mply that evenclosed arc-transtve crculants may only exst amongst normal arc-transtve crculants of order 1 (mod 4). In all other cases an arc-transtve crculant admts an odd automorphsm. Corollary (mod 4). An even-closed arc-transtve crculant s normal and has order For the rest of ths secton we may, n our search for odd automorphsms, therefore restrct ourselves to normal crculants. Let X = Crc(n, S) be a normal arc-transtve crculant of order order 1 (mod 4) and let s S. Then for any s S there must be an automorphsm α of G such that α(s) = s, and so s and s are of the same order. Thus f s s not of order n then Crc(n, S) s not connected. Hence t has at least three components (snce n s not even), and has an automorphsm that fxes all but one component whle rotatng that component, but ths automorphsm does not normalze the regular cyclc subgroup of Aut(X). Ths shows that we may assume that 1 S (note that addtve notaton s used for Z n ). Ths fact s used throughout ths secton. The followng lemma about the acton of the multplcatve group of unts s needed n ths respect. For a postve nteger n we use n p to denote the hghest power of p dvdng n. Lemma Let p be an odd prme, and let k 1 be a postve nteger. Then Z p k, n ts natural acton on Z p k, admts an odd permutaton f and only f k s odd. Proof. By Proposton 2.1 t suffces to consder the Sylow 2-subgroup J of Z p. Snce k Z p s a cyclc group, J s cyclc too. Let α be a generator of J. We clam that α acts k semregularly on Z p k \ {0}. Suppose on the contrary that ths s not the case. Then there exst m N such that α m 1 and α m (x) = x for some x Z p k \ {0}. Ths s equvalent to (α m 1)x 0 (mod p k ).

8 434 Ars Math. Contemp. 10 (2016) The above equaton admts a non-trval soluton f and only f α m 1 s dvsble by p. Suppose that j < k s such that p j α m 1. There exsts A Z such that α m = Ap j + 1 and (A, p) = 1. Snce α m J there exsts s N such that (α m ) 2s 1 (mod p k ). It follows that (Ap j + 1) 2s 1 (mod p k ), and so 2 s ( ) 2 s (Ap j ) 0 (mod p k ). =1 For each > 1 the number ( ) 2 s (Ap j ) s dvsble by p j+1. Consequently, 2 s Ap j s dvsble by p j+1, and so we conclude that p dvdes 2 s A, a contradcton. As clamed above, ths shows that α acts semregularly on Z p k \ {0} wth p 1 (p k p k 1 pk 1 ) 2 p 1 = (1 + p p 1 pk 1 ) (p k p k 1 ) 2 cycles of even length (p k p k 1 ) 2 = (p 1) 2 n ts cycle decomposton (snce α s a generator of J). Snce the party of 1 + p p k 1 depends on whether k s even or odd, t follows that α s an odd permutaton f and only f k s odd. The result follows. Corollary Let p be an odd prme, and let k 1 be a postve nteger such that p k 1 (mod 4). Then a normal arc-transtve crculant X = Cay(Z p k, S) admts an odd automorphsm f and only f k s odd and S contans the Sylow 2-subgroup of Z p k. Proof. Recall that Aut(X) = Z p k S, and thus X admts odd automorphsms f and only f S contans an element gvng rse to an odd permutaton on Z p k (generators of Sylow 2- subgroups of Z p k are odd permutatons on Z p k). The result s thus obtaned by combnng together Proposton 2.1 and Lemma Lemma Let n = p 2k1+1 1 p 2ka+1 a q 2l q 2l b b be a prme decomposton of an odd nteger n, and let Z n = P1 P a Q 1 Q b, where P = Zp 2k +1, {1,..., a}, and Q = Zq 2l Sylow 2-subgroup of P, {1,..., b}. Further, let α and β, respectvely, be generators of the and the Sylow 2-subgroup of Q. Then, for each, we have that α s an odd permutaton on Z n, and β s an even permutaton on Z n. Proof. Observe that each cycle n the cycle decomposton of α P (consdered as a permutaton of Z 2k p +1) s lfted to n/p 2k+1 cycles of the same length n the cycle decomposton of α (when consdered as a permutaton of Z n ). By Lemma 3.12, α s an odd permutaton on Z 2k p +1 for each. Smlarly, β s an even permutaton on Z 2l q for each. Snce n s odd, the result follows. We ntroduce the followng notaton. Let n = p k1 1 pka a be a prme decomposton of a postve nteger n, let Z n = a =1 P, where P = Zp k, and let J(p ) be the Sylow 2-subgroup of P. In the next theorem a necessary and suffcent condton for a normal crculant to be even-closed s gven. One of the mmedate consequences s, for example, that a normal crculant of order n 2, n odd, s even-closed.

9 A. Hujdurovć et al.: Odd automorphsms n vertex-transtve graphs 435 Theorem Let n = p k1 1 pka a be a prme decomposton of a postve nteger n, and let X = Crc(n, S) be a normal arc-transtve crculant on Z n = a =1 P. Then X s even-closed f and only f n 1 (mod 4) and for every α = a =1 α S we have a { 1; f J(p ) α θ (α) 0 (mod 2), where θ (α) = and k s odd 0; otherwse =1. Proof. By Corollary 3.3 for n 1 (mod 4) the graph X admts odd automorphsms. We may therefore assume that n 1 (mod 4). By Lemma 3.14, the exstence of odd automorphsms n X depends solely on the party of the exponents k and the contanment of the generators of the correspondng Sylow 2-subgroups n S, and the result follows. (Recall, that we are usng the assumpton that 1 S and the fact that n a normal arctranstve crculant, every element of S s conjugate, so every α S s odd f and only f X has an odd automorphsm.) Combned together wth Corollary 3.11 and Theorem 3.15 we have the followng characterzaton of even-closed arc-transtve crculants. Theorem Let X be an even-closed arc-transtve crculant of order n and let n = p k1 1 pka a be a prme decomposton of n. Then X s a normal crculant X = Crc(n, S) on Z n = a =1 P, n 1 (mod 4) and for every α = a =1 α S we have a { 1; f J(p ) α θ (α) 0 (mod 2), where θ (α) = and k s odd 0; otherwse =1. 4 Vertex-transtve graphs It s known that every fnte transtve permutaton group contans a fxed-pont-free element of prme power order (see [7, Theorem 1]), but not necessarly a fxed-pont-free element of prme order and, hence, a semregular element (see for nstance [2, 7]). In 1981 the thrd author asked f every vertex-transtve dgraph wth at least two vertces admts a semregular automorphsm (see [17, Problem 2.4]). Despte consderable efforts by varous mathematcans the problem remans open, wth the class of vertex-transtve graphs havng a solvable automorphsm group beng the man obstacle. The most recent result on the subject s due to Verret [24] who proved that every arc-transtve graph of valency 8 has a semregular automorphsm, whch was the smallest open valency for arc-transtve graphs (see [5, 8, 20] and [12] for an overvew of the status of ths problem). Whle the exstence of such automorphsms n certan vertex-transtve graphs has proved to be an mportant buldng block n obtanng at least partal solutons n many open problems n algebrac graph theory, such as for example the hamltoncty problem (see [11, 13, 16]), the connecton to the even/odd problem s straghtforward. Proposton 4.1. An even-closed vertex-transtve graph does not have even order semregular automorphsms wth an odd number of orbts. Ths suggest that n a search for odd automorphsms a specal attenton should be gven to semregular automoprhsms of even order. Furthermore, for those classes of vertex-transtve graphs for whch a complete classfcaton (together wth the correspondng automorphsm groups) exsts, the answer to

10 436 Ars Math. Contemp. 10 (2016) Problem 1.1 s, at least mplctly, avalable rght there n the classfcaton. Such s, for example, the case of vertex-transtve graphs of order a product of two prmes, see [4, 6, 18, 19, 22], and the case of vertex-transtve graphs whch are graph truncatons, see [1]. The hard work needed to complete these classfcatons suggest that the even/odd queston s by no means an easy one. Let us consder, for example, the class of all vertextranstve graphs of order 2p, p a prme. In the completon of the classfcaton of these graphs, the classfcaton of fnte smple groups s an essental ngredent n handlng the case of prmtve automorphsm groups. We know, by ths classfcaton, that the Petersen graph and ts complement are the only such graphs wth a prmtve automorphsm group. Of course, they both admt odd automorphsms. As for mprmtve automorphsm groups, t all depends on the arthmetc of p. When p 3 (mod 4), the graphs are necessarly Cayley graphs (of dhedral groups) and hence must admt odd automorphsms. (Namely, reflectons nterchangng the two orbts of the rotaton n the dhedral group are odd automorphsms.) When p 1 (mod 4), then t follows by the classfcaton of these graphs [17] that there s an automorphsm of order 2 k, k 1, nterchangng the two blocks of mprmtvty of sze p, havng one orbt of sze 2 and 2(p 1)/2 k orbts of sze 2 k, thus an odd number of orbts n total (snce 2 k dvdes p 1). We have thus shown that every vertex-transtve graph of order twce a prme number admts an odd automorphsm. However, no classfcaton of fnte smple groups free proof of the above fact s known to us. In concluson we would lke to make the followng comment wth regards to cubc vertex-transtve graphs. Recall that the class of cubc vertex-transtve graphs decomposes nto three subclasses dependng on the number of orbts of the vertex-stablzer on the set of neghbors of a vertex. These subclasses are arc-transtve graphs (one orbt), graphs wth vertex-stablzers beng 2-groups (two orbts) and GRR graphs (three orbts), see [21]. (Note that there are two types of cubc GRR graphs, those wth connectng set consstng of three nvolutons and those wth connectng set consstng of an nvoluton, a non-nvoluton and ts nverse, see [3].) For the frst and thrd subclass the answer to Problem 1.1 s gven n [14] and Proposton 3.1, respectvely, whle the problem s stll open for the second subclass. Examples gven n Secton 1 (see also Fgure 2) show, however, that ths second subclass contans nfntely many even-closed graphs as well as nfntely many graphs admttng odd automorphsms. Problem 4.2. Classfy cubc vertex-transtve graphs wth vertex-stablzers beng 2-groups that admt odd automorphsms. References [1] B. Alspach and E. Dobson, On automorphsm groups of graph truncatons, Ars Math. Contemp. 8 (2015), [2] P. J. Cameron, M. Gudc, W. M. Kantor, G. A. Jones, M. H. Kln, D. Marušč and L. A. Nowtz, Transtve permutaton groups wthout semregular subgroups, J. London Math. Soc. 66 (2002), [3] M. D. E. Conder, T. Psansk and A. Žtnk, Vertex-transtve graphs and ther arc-types, arxv: [4] E. Dobson, Automorphsm groups of metacrculant graphs of order a product of two dstnct prmes, Combn. Probab. Comput. 15 (2006), [5] E. Dobson, A. Malnč, D. Marušč and L. A. Nowtz, Semregular automorphsms of vertex-transtve graphs of certan valences, J. Combn. Theory, Ser. B 97 (2007), [6] E. Dobson and D. Wtte, Transtve permutaton groups of prme-squared degree, J. Algebrac Combn. 16 (2002),

11 A. Hujdurovć et al.: Odd automorphsms n vertex-transtve graphs 437 [7] B. Fen, W. M. Kantor and M. Schacher, Relatve Brauer groups II, J. Rene Angew. Mat. 328 (1981), [8] M. Gudc and G. Verret, Semregular automorphsms n arc-transtve graphs of valency 2p, n preparaton. [9] C. D. Godsl, On the full automorphsm group of a graph, Combn. 1 (1981), [10] I. Kovács, Classfyng arc-transtve crculants, J. Algebr. Combn. 20 (2004), [11] K. Kutnar and D. Marušč, Hamltoncty of vertex-transtve graphs of order 4p, European J. Combn. 29 (2008) [12] K. Kutnar and D. Marušč, Recent trends and future drectons n vertex-transtve graphs, Ars Math. Contemp. 1 (2008), [13] K. Kutnar and P. Šparl, Hamlton paths and cycles n vertex-transtve graphs of order 6p, Dscrete Math. 309 (2009) [14] K. Kutnar and D. Marušč, Cubc symmetrc graphs va odd automorphsms, manuscrpt. [15] C. H. L, Permutaton groups wth a cyclc regular subgroup and arc-transtve crculants, J. Algebr. Combn. 21 (2005), [16] L. Lovász, Combnatoral structures and ther applcatons, (Proc. Calgary Internat. Conf., Calgary, Alberta, 1969), pp , Problem 11, Gordon and Breach, New York, [17] D. Marušč, On vertex symmetrc dgraphs, Dscrete Math. 36 (1981), [18] D. Marušč and R. Scapellato, Characterzng vertex-transtve pq-graphs wth mprmtve automorphsm subgroups, J. Graph Theory 16 (1992), [19] D. Marušč and R. Scapellato, Classfyng vertex-transtve graphs whose order s a product of two prmes, Combnatorcs 14 (1994), [20] D. Marušč and R. Scapellato, Permutaton groups, vertex-transtve dgraphs and semregular automorphsms, European J. Combn. 19 (1998), [21] P. Potočnk, P. Spga and G. Verret, Cubc vertex-transtve graphs on up to 1280 vertces, J. Symbolc Computaton 50 (2013), [22] C. E. Praeger and M. Y. Xu, Vertex-prmtve graphs of order a product of two dstnct prmes, J. Combn. Theory, Ser. B 59 (1993), [23] G. Sabduss, On a class of fxed-pont-free graphs. Proc. Amer. Math. Soc. 9 (1958), [24] G. Verret, Arc-transtve graphs of valency 8 have a semregular automorphsm, Ars Math. Contemp. 8 (2015), [25] M. Y. Xu, Automorphsm groups and somorphsms of Cayley dgraphs, Dscrete Math. 182 (1998),