A graph obtained by deleting a Hamming code of length \(n= 2^r - 1\) from a n-cube \(Q_n\) is called as a Hamming shell. It is well known that a Hamming shell is regular, vertex-transitive, edge-transitive and distance preserving (Borges and Dejter in J Comb Math Comb Comput 20:161–173, 1996; Dejter in Discrete Math 124:55–66, 1994; J Comb Des 5:301–309, 1997; Discrete Math 261:177–187, 2003). Moreover, it is Hamiltonian (Gregor and Skrekovski in Discrete Math Theor Comput Sci 11(1):187–200, 2009) and connected (Duckworth et al. in Appl Math Lett 14:801–804, 2001). In this paper, we prove that a Hamming shell is edge-bipancyclic and \((n-1)\)-connected.
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The first author gratefully acknowledges the Department of Science and Technology, New Delhi, India for the award of Women Scientist Scheme (SR/WOS-A/PM-79/2016) for research in Basic/Applied Sciences. The authors would like to express their gratitude to the anonymous referees for their kind suggestions and corrections that helped to improve the original manuscript.
Borges, J., Dejter, I.J.: On perfect dominating sets in hypercubes and their complements. J. Comb. Math. Comb. Comput. 20, 161–173 (1996)MathSciNetzbMATHGoogle Scholar