Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 215–223 On automorphism groups of graph truncations Brian Alspach School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia Edward Dobson Departmentment of Mathematics and Statisitics Mississippi State University Mississippi State, MS 39762 USA and UP IAM University of Primorska Muzejeska trg 2, 6000 Koper, Slovenia Received 11 May 2014, accepted 13 November 2014, published online 17 December 2014 Abstract It is well known that the Petersen graph, the Coxeter graph, as well as the graphs ob- tained from these two graphs by replacing each vertex with a triangle, are trivalent vertex- transitive graphs without Hamilton cycles, and are indeed the only known connected vertex- transitive graphs of valency at least two without Hamilton cycles. It is known by many that the replacement of a vertex with a triangle in a trivalent vertex-transitive graph results in a vertex-transitive graph if and only if the original graph is also arc-transitive. In this paper, we generalize this notion to t-regular graphs Γ and replace each vertex with a complete graph K t on t vertices. We determine necessary and sufficient conditions for T (Γ) to be hamiltonian, show Aut(T (Γ)) ∼ = Aut(Γ), as well as show that if Γ is vertex-transitive, then T (Γ) is vertex-transitive if and only if Γ is arc-transitive. Finally, in the case where t is prime we determine necessary and sufficient conditions for T (Γ) to be isomorphic to a Cayley graph as well as an additional necessary and sufficient condition for T (Γ) to be vertex-transitive. Keywords: Truncation, automorphism group, Cayley graph, Hamiltonian. Math. Subj. Class.: 05C25 E-mail addresses: brian.alspach@newcastle.edu.au (Brian Alspach), dobson@math.msstate.edu (Edward Dobson) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/