- Title
- On automorphism groups of graph truncations
- Creator
- Alspach, Brian; Dobson, Edward
- Relation
- Ars Mathematica Contemporanea Vol. 8, Issue 1, p. 215-223
- Relation
- http://amc-journal.eu/index.php/amc/article/view/665
- Publisher
- Society of Mathematicians, Physicists and Astronomers
- Resource Type
- journal article
- Date
- 2015
- Description
- 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.
- Subject
- Truncation; automorphism group; Cayley graph; Hamiltonian
- Identifier
- http://hdl.handle.net/1959.13/1339599
- Identifier
- uon:28295
- Identifier
- ISSN:1855-3966
- Rights
- This work is licensed under http://creativecommons.org/licenses/by/3.0/.
- Language
- eng
- Full Text
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