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• W919683003 abstract "In the first part of the thesis we define an automorphism φn for each star graphStn of degree n − 1, which yields permutations of labels for the edges of Stntaken from the set of integers {1, . . . , bn/2c}. By decomposing these permutationsinto permutation cycles, we are able to identify edge-disjoint Hamilton cyclesthat are automorphic images of a known two-labelled Hamilton cycle H1 2(n)in Stn. Our main result is an improvement from the existing lower bound ofbϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known numberof edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, theimprovement is from bn/8c to bn/5c. We extend this result to the cases when nis the power of a prime other than 3 and 7.The second part of the thesis studies ‘symmetric’ collections of edge-disjointHamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) undergeneral label-mapping automorphisms. We show that, for all even n, there existsa symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannothave symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus,Stn is not symmetrically Hamilton decomposable if n is not prime. We also givecases of even n, in terms of Carmichael’s reduced totient function λ, for which‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generatedfrom H1 2(n) by a single automorphism, can and cannot attain the optimumbound bϕ(n)/2c for symmetric collections. In particular, we show that if n is apower of 2, then Stn has a spanning subgraph with more than half of the edgesof Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remainsan open problem as to whether the bϕ(n)/2c can be achieved for symmetriccollections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound isachievable and optimal for strongly symmetric collections.The search for edge-disjoint Hamilton cycles in star graphs is important for thedesign of interconnection network topologies in computer science. All our resultsimprove on the known bounds for numbers of any kind of edge-disjoint Hamiltoncycles in star graphs." @default.
• W919683003 created "2016-06-24" @default.
• W919683003 creator A5025894616 @default.
• W919683003 date "2015-01-01" @default.
• W919683003 modified "2023-09-22" @default.
• W919683003 title "Automorphisms generating disjoint Hamilton cycles in star graphs" @default.
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