The Existence and Construction of (K5∖e)-Designs of Orders 27, 135, 162, and 216
The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K5∖e has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a (K27,K5∖e)‐des...
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| Published in | Journal of combinatorial designs Vol. 21; no. 7; pp. 280 - 302 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Blackwell Publishing Ltd
01.07.2013
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1063-8539 1520-6610 |
| DOI | 10.1002/jcd.21340 |
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| Abstract | The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K5∖e has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a (K27,K5∖e)‐design. I show that |Aut(Γ)| divides 2k3 for some k≥0 and that Sym(3)≰Aut(Γ). I construct (K27,K5∖e)‐designs by prescribing Z6 as an automorphism group, and show that up to isomorphism there are exactly 24 (K27,K5∖e)‐designs with Z6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z6. Finally, the existence of (K5∖e)‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864. |
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| AbstractList | The problem of the existence of a decomposition of the complete graph K n into disjoint copies of K 5 e has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let [Gamma] be a ( K 27 , K 5 e )-design. I show that | A u t ( [Gamma] ) | divides 2k3 for some k ≥ 0 and that S y m ( 3 ) ≤ A u t ( [Gamma] ). I construct ( K 27 , K 5 e )-designs by prescribing Z 6 as an automorphism group, and show that up to isomorphism there are exactly 24 ( K 27 , K 5 e )-designs with Z 6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z 6. Finally, the existence of ( K 5 e )-designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851-864. [PUBLICATION ABSTRACT] The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n , except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a ‐design. I show that divides 2 k 3 for some and that . I construct ‐designs by prescribing as an automorphism group, and show that up to isomorphism there are exactly 24 ‐designs with as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed . Finally, the existence of ‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864. The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K5∖e has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a (K27,K5∖e)‐design. I show that |Aut(Γ)| divides 2k3 for some k≥0 and that Sym(3)≰Aut(Γ). I construct (K27,K5∖e)‐designs by prescribing Z6 as an automorphism group, and show that up to isomorphism there are exactly 24 (K27,K5∖e)‐designs with Z6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z6. Finally, the existence of (K5∖e)‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864. |
| Author | Kolotoğlu, Emre |
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| References | G. Ge and A. C. H. Ling, On the existence of (K5∖e)-designs with application to optical networks, SIAM J Discrete Math 21(4) (2007), 851-864. Q. Li and Y. Chang, A few more (Kv,K5−e)-designs, Bull Inst Combin Appl 45 (2005) 11-16. S. G. Hartke, P. R. J. Östergård, D. Bryant, and S. I. El-Zanati, The nonexistence of a (K6−e)-decomposition of the complete graph K29, J Combin Des 18 (2010), 94-104. D. Dor and M. Tarsi, Graph decomposition is NP-complete: A proof of Holyer's conjecture, SIAM J Comput 26 (1997), 1166-1187. P. Adams, D. Bryant, and M. Buchanan, A survey on the existence of G-designs, J Combin Des 16 (2008), 373-410. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 2nd edition, Cambridge University Press, Cambridge, UK, 1999. J. C. Bermond, C. Huang, A. Rosa, and D. Sotteau, Decomposition of complete graphs into isomorphic subgraphswith five vertices, Ars Combin 10 (1980), 211-254. 2007 1996 2007; 21 2010; 18 1997; 26 2005; 45 2008; 16 1980; 10 1999 e_1_2_7_5_1 e_1_2_7_3_1 Hartke S. G. (e_1_2_7_9_1) 2010; 18 Bryant D. (e_1_2_7_6_1) 2007 e_1_2_7_8_1 Abel R. J. R. (e_1_2_7_2_1) 2007 e_1_2_7_7_1 Heinrich K. (e_1_2_7_10_1) 1996 Li Q. (e_1_2_7_11_1) 2005; 45 Bermond J. C. (e_1_2_7_4_1) 1980; 10 |
| References_xml | – reference: Q. Li and Y. Chang, A few more (Kv,K5−e)-designs, Bull Inst Combin Appl 45 (2005) 11-16. – reference: G. Ge and A. C. H. Ling, On the existence of (K5∖e)-designs with application to optical networks, SIAM J Discrete Math 21(4) (2007), 851-864. – reference: S. G. Hartke, P. R. J. Östergård, D. Bryant, and S. I. El-Zanati, The nonexistence of a (K6−e)-decomposition of the complete graph K29, J Combin Des 18 (2010), 94-104. – reference: P. Adams, D. Bryant, and M. Buchanan, A survey on the existence of G-designs, J Combin Des 16 (2008), 373-410. – reference: J. C. Bermond, C. Huang, A. Rosa, and D. Sotteau, Decomposition of complete graphs into isomorphic subgraphswith five vertices, Ars Combin 10 (1980), 211-254. – reference: D. Dor and M. Tarsi, Graph decomposition is NP-complete: A proof of Holyer's conjecture, SIAM J Comput 26 (1997), 1166-1187. – reference: T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 2nd edition, Cambridge University Press, Cambridge, UK, 1999. – volume: 45 start-page: 11 year: 2005 end-page: 16 article-title: A few more ‐designs publication-title: Bull Inst Combin Appl – start-page: 160 year: 2007 end-page: 193 – volume: 10 start-page: 211 year: 1980 end-page: 254 article-title: Decomposition of complete graphs into isomorphic subgraphswith five vertices publication-title: Ars Combin – volume: 26 start-page: 1166 year: 1997 end-page: 1187 article-title: Graph decomposition is NP‐complete: A proof of Holyer's conjecture publication-title: SIAM J Comput – start-page: 361 year: 1996 end-page: 366 – start-page: 477 year: 2007 end-page: 486 – volume: 21 start-page: 851 issue: 4 year: 2007 end-page: 864 article-title: On the existence of ‐designs with application to optical networks publication-title: SIAM J Discrete Math – volume: 18 start-page: 94 year: 2010 end-page: 104 article-title: The nonexistence of a ‐decomposition of the complete graph publication-title: J Combin Des – volume: 16 start-page: 373 year: 2008 end-page: 410 article-title: A survey on the existence of ‐designs publication-title: J Combin Des – year: 1999 – start-page: 160 volume-title: Handbook of Combinatorial Designs year: 2007 ident: e_1_2_7_2_1 – ident: e_1_2_7_5_1 doi: 10.1017/CBO9780511549533 – ident: e_1_2_7_7_1 doi: 10.1137/S0097539792229507 – start-page: 361 volume-title: The CRC Handbook of Combinatorial Designs year: 1996 ident: e_1_2_7_10_1 – ident: e_1_2_7_3_1 doi: 10.1002/jcd.20170 – volume: 18 start-page: 94 year: 2010 ident: e_1_2_7_9_1 article-title: The nonexistence of a ‐decomposition of the complete graph K 29 publication-title: J Combin Des doi: 10.1002/jcd.20226 – ident: e_1_2_7_8_1 doi: 10.1137/060660084 – volume: 45 start-page: 11 year: 2005 ident: e_1_2_7_11_1 article-title: A few more ‐designs publication-title: Bull Inst Combin Appl – volume: 10 start-page: 211 year: 1980 ident: e_1_2_7_4_1 article-title: Decomposition of complete graphs into isomorphic subgraphswith five vertices publication-title: Ars Combin – start-page: 477 volume-title: Handbook of Combinatorial Designs year: 2007 ident: e_1_2_7_6_1 |
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| Snippet | The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K5∖e has been solved for all admissible orders n, except for... The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n , except for 27, 36,... The problem of the existence of a decomposition of the complete graph K n into disjoint copies of K 5 e has been solved for all admissible orders n, except for... |
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| SubjectTerms | (K5∖e)-design 05C51 graph decomposition isomorph rejection 2010 Mathematics Subject Classifications: 05B30 isomorph rejection 2010 Mathematics Subject Classifications: 05B30, 05C51 Studies |
| Title | The Existence and Construction of (K5∖e)-Designs of Orders 27, 135, 162, and 216 |
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