Distance in cayley graphs on permutation groups generated by $k$ $m$-Cycles

Document Type: Research Paper

Authors

Iran University of Science and Technology

Abstract

‎‎In this paper‎, ‎we extend upon the results of B‎. ‎Suceav{\u{a}} and R‎. ‎Stong [Amer‎. ‎Math‎. ‎Monthly‎, ‎110 (2003) 162--162]‎, ‎which they computed the minimum number of 3-cycles needed to generate an even permutation‎.
‎Let $\Omega^n_{k,m}$ be the set of all permutations of the form $c_1 c_2 \cdots c_k$‎ ‎where $c_i$'s are arbitrary $m$-cycles in $S_n$‎. ‎Suppose that $\Gamma^n_{k,m}$ be the Cayley graph on subgroup of $S_n$ generated by all permutations‎ ‎in $\Omega^n_{k,m}$‎. ‎We find a shortest path joining identity and any vertex of $\Gamma^n_{k,m}$‎, ‎for arbitrary natural number $k$‎, ‎and $m=2‎ , ‎\‎, ‎3,\‎, ‎4$‎. ‎Also‎, ‎we calculate the diameter of these Cayley graphs‎. ‎As an application‎, ‎we present an algorithm for finding a short expression of a permutation as products of given permutations‎.

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