On automorphism groups of graph truncations
Ars Mathematica Contemporanea, Tome 8 (2015) no. 1, pp. 215-223.

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It is well known that the Petersen graph, the Coxeter graph, as well as the graphs obtained 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 Kt 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.
DOI : 10.26493/1855-3974.665.4b6
Mots-clés : Truncation, automorphism group, Cayley graph, Hamiltonian. 
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Brian Alspach; Edward Dobson. On automorphism groups of graph truncations. Ars Mathematica Contemporanea, Tome 8 (2015) no. 1, pp. 215-223. doi : 10.26493/1855-3974.665.4b6. https://geodesic-test.mathdoc.fr/articles/10.26493/1855-3974.665.4b6/

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