Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 427–437 Odd automorphisms in vertex-transitive graphs Ademir Hujdurovi´ c ∗ , Klavdija Kutnar † University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaˇ ska 8, 6000 Koper, Slovenia Dragan Maruˇ siˇ c ‡ University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaˇ ska 8, 6000 Koper, Slovenia IMFM, Jadranska 19, 1000 Ljubljana, Slovenia Received 24 February 2016, accepted 10 July 2016, published online 25 July 2016 Abstract An automorphism of a graph is said to be even/odd if it acts on the set of vertices as an even/odd permutation. In this article we pose the problem of determining which vertex-transitive graphs admit odd automorphisms. Partial results for certain classes of vertex-transitive graphs, in particular for Cayley graphs, are given. As a consequence, a characterization of arc-transitive circulants without odd automorphisms is obtained. Keywords: Graph, vertex-transitive, automorphism group, even permutation, odd permutation. Math. Subj. Class.: 20B25, 05C25 1 Introduction Apart from being a rich source of interesting mathematical objects in their own right, vertex-transitive graphs provide a perfect platform for investigating structural properties of transitive permutation groups from a purely combinatorial viewpoint. The recent outburst * This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0032, N1-0038, and J1-7051). † This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0032, N1-0038, J1-6720, J1-6743, and J1-7051), in part by WoodWisdom-Net+, W 3 B, and in part by NSFC project 11561021. ‡ This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0032, N1-0038, J1-5433, J1-6720, and J1-7051), and in part by H2020 Teaming InnoRenew CoE. E-mail address: ademir.hujdurovic@upr.si (Ademir Hujdurovi´ c), klavdija.kutnar@upr.si (Klavdija Kutnar)dragan.marusic@upr.si (Dragan Maruˇ siˇ c) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/