Connected Covering Numbers
A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results...
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| Published in | Journal of combinatorial designs Vol. 23; no. 12; pp. 534 - 549 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Blackwell Publishing Ltd
01.12.2015
Wiley Subscription Services, Inc Wiley |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1063-8539 1520-6610 |
| DOI | 10.1002/jcd.21425 |
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| Abstract | A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with Turán systems. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas. |
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| AbstractList | A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with Turán systems. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas. |
| Author | Chappelon, Jonathan Ramírez Alfonsín, Jorge Luis Montejano, Luis Pedro Knauer, Kolja |
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| Keywords | uniform oriented matroid Turán-system covering design |
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| References | D. Forge, J. L. Ramírez Alfonsín, and H. Yeun, Disconnected coverings for oriented matroids via simultaneous mutations, Discrete Math 258 (2002), 353-359. D. Forge and J. L. Ramírez Alfonsín, Connected coverings and an application to oriented matroids, Discrete Math 187 (1998), 109-121. W. H. Mills, A covering of triples by quadruples, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), Congr Numer 33 (1981), 253-260. P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat Fiz Lapok 48 (1941), 436-452. D. Applegate, E. M. Rains, and N. J. A. Sloane, On asymmetric coverings and covering numbers, J Combin Des 11 (2003), 218-228. J. Schönheim, On coverings, Pacific J Math 14 (1964), 1405-1411. Y. O. Hamidoune and M. Las Vergnas, Directed switching games on graphs and matroids, J Combin Theory Ser B 40 (1986), 237-269. V. Rödl, On a packing and covering problem, Eur J Combin 6 (1985), 69-78. A. F. Sidorenko, The method of quadratic forms in a combinatorial problem of Turán, Vestnik Moskov. Univ. Ser. I Mat Mekh 76 (1982), 3-6. A. V. Kostochka, A class of constructions for Turán's (3,4)-problem, Combinatorica 2 (1982), 187-192. W. Mantel, Problem 28, Wiskundige Opgaven 10 (1907), 320. M. K. Fort Jr. and G. A. Hedlund, Minimal coverings of pairs by triples, Pacific J Math 8 (1958), 709-719. N. N. Kuzjurin, Explicit constructions of Rödl's asymptotically good packings and coverings, Combin Probab Comput 9 (2000), 265-276. G. Katona, T. Nemetz, and M. Simonovits, On a problem of Turán in the theory of graphs, Mat Lapok 15 (1964), 228-238. P. Turán, Research problems, Magyar Tud Akad Mat Kut Int Közl 6 (1961), 417-423. L. Ji, An improvement on covering triples by quadruples, J Combin Des 16 (2008), 231-243. A. F. Sidorenko, What we know and what we do not know about Turán numbers, Graphs Combin 11 (1995), 179-199. D. M. Gordon, O. Patashnik, and G. Kuperberg, New constructions for covering designs, J Combin Des 3 (1995), 269-284. 1986; 40 1941; 48 1982; 2 1961; 6 2000; 9 2008; 16 1982; 76 1995; 11 1985; 6 2002; 258 1958; 8 1974 1907; 10 1964; 14 1964; 15 1998; 187 1995; 3 2003; 11 1999 1981; 33 Katona (10.1002/jcd.21425-BIB0009|jcd21425-cit-0009) 1964; 15 Hamidoune (10.1002/jcd.21425-BIB0007|jcd21425-cit-0007) 1986; 40 Forge (10.1002/jcd.21425-BIB0004|jcd21425-cit-0004) 2002; 258 Mantel (10.1002/jcd.21425-BIB0013|jcd21425-cit-0013) 1907; 10 Mills (10.1002/jcd.21425-BIB0015|jcd21425-cit-0015) 1981; 33 Schönheim (10.1002/jcd.21425-BIB0017|jcd21425-cit-0017) 1964; 14 10.1002/jcd.21425-BIB0002|jcd21425-cit-0002 Ji (10.1002/jcd.21425-BIB0008|jcd21425-cit-0008) 2008; 16 10.1002/jcd.21425-BIB0014|jcd21425-cit-0014 Fort (10.1002/jcd.21425-BIB0005|jcd21425-cit-0005) 1958; 8 Turán (10.1002/jcd.21425-BIB0021|jcd21425-cit-0021) 1961; 6 Kuzjurin (10.1002/jcd.21425-BIB0012|jcd21425-cit-0012) 2000; 9 Rödl (10.1002/jcd.21425-BIB0016|jcd21425-cit-0016) 1985; 6 Forge (10.1002/jcd.21425-BIB0003|jcd21425-cit-0003) 1998; 187 Sidorenko (10.1002/jcd.21425-BIB0019|jcd21425-cit-0019) 1995; 11 Turán (10.1002/jcd.21425-BIB0020|jcd21425-cit-0020) 1941; 48 Applegate (10.1002/jcd.21425-BIB0001|jcd21425-cit-0001) 2003; 11 Gordon (10.1002/jcd.21425-BIB0006|jcd21425-cit-0006) 1995; 3 Kostochka (10.1002/jcd.21425-BIB0011|jcd21425-cit-0011) 1982; 2 10.1002/jcd.21425-BIB0010|jcd21425-cit-0010 Sidorenko (10.1002/jcd.21425-BIB0018|jcd21425-cit-0018) 1982; 76 |
| References_xml | – reference: V. Rödl, On a packing and covering problem, Eur J Combin 6 (1985), 69-78. – reference: A. F. Sidorenko, What we know and what we do not know about Turán numbers, Graphs Combin 11 (1995), 179-199. – reference: G. Katona, T. Nemetz, and M. Simonovits, On a problem of Turán in the theory of graphs, Mat Lapok 15 (1964), 228-238. – reference: W. Mantel, Problem 28, Wiskundige Opgaven 10 (1907), 320. – reference: D. Applegate, E. M. Rains, and N. J. A. Sloane, On asymmetric coverings and covering numbers, J Combin Des 11 (2003), 218-228. – reference: A. F. Sidorenko, The method of quadratic forms in a combinatorial problem of Turán, Vestnik Moskov. Univ. Ser. I Mat Mekh 76 (1982), 3-6. – reference: N. N. Kuzjurin, Explicit constructions of Rödl's asymptotically good packings and coverings, Combin Probab Comput 9 (2000), 265-276. – reference: W. H. Mills, A covering of triples by quadruples, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), Congr Numer 33 (1981), 253-260. – reference: J. Schönheim, On coverings, Pacific J Math 14 (1964), 1405-1411. – reference: D. M. Gordon, O. Patashnik, and G. Kuperberg, New constructions for covering designs, J Combin Des 3 (1995), 269-284. – reference: D. Forge and J. L. Ramírez Alfonsín, Connected coverings and an application to oriented matroids, Discrete Math 187 (1998), 109-121. – reference: P. Turán, Research problems, Magyar Tud Akad Mat Kut Int Közl 6 (1961), 417-423. – reference: L. Ji, An improvement on covering triples by quadruples, J Combin Des 16 (2008), 231-243. – reference: D. Forge, J. L. Ramírez Alfonsín, and H. Yeun, Disconnected coverings for oriented matroids via simultaneous mutations, Discrete Math 258 (2002), 353-359. – reference: A. V. Kostochka, A class of constructions for Turán's (3,4)-problem, Combinatorica 2 (1982), 187-192. – reference: P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat Fiz Lapok 48 (1941), 436-452. – reference: Y. O. Hamidoune and M. Las Vergnas, Directed switching games on graphs and matroids, J Combin Theory Ser B 40 (1986), 237-269. – reference: M. K. Fort Jr. and G. A. Hedlund, Minimal coverings of pairs by triples, Pacific J Math 8 (1958), 709-719. – volume: 11 start-page: 179 year: 1995 end-page: 199 article-title: What we know and what we do not know about Turán numbers publication-title: Graphs Combin – volume: 10 start-page: 320 year: 1907 article-title: Problem 28 publication-title: Wiskundige Opgaven – volume: 3 start-page: 269 year: 1995 end-page: 284 article-title: New constructions for covering designs publication-title: J Combin Des – volume: 14 start-page: 1405 year: 1964 end-page: 1411 article-title: On coverings publication-title: Pacific J Math – volume: 40 start-page: 237 year: 1986 end-page: 269 article-title: Directed switching games on graphs and matroids publication-title: J Combin Theory Ser B – volume: 48 start-page: 436 year: 1941 end-page: 452 article-title: Eine Extremalaufgabe aus der Graphentheorie publication-title: Mat Fiz Lapok – volume: 187 start-page: 109 year: 1998 end-page: 121 article-title: Connected coverings and an application to oriented matroids publication-title: Discrete Math – volume: 16 start-page: 231 year: 2008 end-page: 243 article-title: An improvement on covering triples by quadruples publication-title: J Combin Des – volume: 2 start-page: 187 year: 1982 end-page: 192 article-title: A class of constructions for Turán's (3,4)‐problem publication-title: Combinatorica – volume: 15 start-page: 228 year: 1964 end-page: 238 article-title: On a problem of Turán in the theory of graphs publication-title: Mat Lapok – volume: 11 start-page: 218 year: 2003 end-page: 228 article-title: On asymmetric coverings and covering numbers publication-title: J Combin Des – volume: 33 start-page: 253 year: 1981 end-page: 260 article-title: A covering of triples by quadruples, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981) publication-title: Congr Numer – volume: 6 start-page: 417 year: 1961 end-page: 423 article-title: Research problems publication-title: Magyar Tud Akad Mat Kut Int Közl – start-page: 17 – volume: 6 start-page: 69 year: 1985 end-page: 78 article-title: On a packing and covering problem publication-title: Eur J Combin – volume: 8 start-page: 709 year: 1958 end-page: 719 article-title: Minimal coverings of pairs by triples publication-title: Pacific J Math – start-page: 563 year: 1974 end-page: 581 – volume: 9 start-page: 265 year: 2000 end-page: 276 article-title: Explicit constructions of Rödl's asymptotically good packings and coverings publication-title: Combin Probab Comput – volume: 76 start-page: 3 year: 1982 end-page: 6 article-title: The method of quadratic forms in a combinatorial problem of Turán publication-title: Vestnik Moskov. Univ. Ser. I Mat Mekh – year: 1999 – volume: 258 start-page: 353 year: 2002 end-page: 359 article-title: Disconnected coverings for oriented matroids via simultaneous mutations publication-title: Discrete Math – ident: 10.1002/jcd.21425-BIB0010|jcd21425-cit-0010 – ident: 10.1002/jcd.21425-BIB0014|jcd21425-cit-0014 – volume: 6 start-page: 417 year: 1961 ident: 10.1002/jcd.21425-BIB0021|jcd21425-cit-0021 article-title: Research problems publication-title: Magyar Tud Akad Mat Kut Int Közl – volume: 2 start-page: 187 year: 1982 ident: 10.1002/jcd.21425-BIB0011|jcd21425-cit-0011 article-title: A class of constructions for Turán's (3,4)-problem publication-title: Combinatorica doi: 10.1007/BF02579317 – volume: 14 start-page: 1405 year: 1964 ident: 10.1002/jcd.21425-BIB0017|jcd21425-cit-0017 article-title: On coverings publication-title: Pacific J Math doi: 10.2140/pjm.1964.14.1405 – volume: 15 start-page: 228 year: 1964 ident: 10.1002/jcd.21425-BIB0009|jcd21425-cit-0009 article-title: On a problem of Turán in the theory of graphs publication-title: Mat Lapok – volume: 258 start-page: 353 year: 2002 ident: 10.1002/jcd.21425-BIB0004|jcd21425-cit-0004 article-title: Disconnected coverings for oriented matroids via simultaneous mutations publication-title: Discrete Math doi: 10.1016/S0012-365X(02)00536-8 – volume: 6 start-page: 69 year: 1985 ident: 10.1002/jcd.21425-BIB0016|jcd21425-cit-0016 article-title: On a packing and covering problem publication-title: Eur J Combin doi: 10.1016/S0195-6698(85)80023-8 – volume: 8 start-page: 709 year: 1958 ident: 10.1002/jcd.21425-BIB0005|jcd21425-cit-0005 article-title: Minimal coverings of pairs by triples publication-title: Pacific J Math doi: 10.2140/pjm.1958.8.709 – volume: 187 start-page: 109 year: 1998 ident: 10.1002/jcd.21425-BIB0003|jcd21425-cit-0003 article-title: Connected coverings and an application to oriented matroids publication-title: Discrete Math doi: 10.1016/S0012-365X(97)00246-X – volume: 10 start-page: 320 year: 1907 ident: 10.1002/jcd.21425-BIB0013|jcd21425-cit-0013 article-title: Problem 28 publication-title: Wiskundige Opgaven – volume: 11 start-page: 218 year: 2003 ident: 10.1002/jcd.21425-BIB0001|jcd21425-cit-0001 article-title: On asymmetric coverings and covering numbers publication-title: J Combin Des doi: 10.1002/jcd.10022 – volume: 40 start-page: 237 year: 1986 ident: 10.1002/jcd.21425-BIB0007|jcd21425-cit-0007 article-title: Directed switching games on graphs and matroids publication-title: J Combin Theory Ser B doi: 10.1016/0095-8956(86)90083-3 – volume: 48 start-page: 436 year: 1941 ident: 10.1002/jcd.21425-BIB0020|jcd21425-cit-0020 article-title: Eine Extremalaufgabe aus der Graphentheorie publication-title: Mat Fiz Lapok – volume: 16 start-page: 231 year: 2008 ident: 10.1002/jcd.21425-BIB0008|jcd21425-cit-0008 article-title: An improvement on covering triples by quadruples publication-title: J Combin Des doi: 10.1002/jcd.20156 – volume: 33 start-page: 253 year: 1981 ident: 10.1002/jcd.21425-BIB0015|jcd21425-cit-0015 article-title: A covering of triples by quadruples, Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981) publication-title: Congr Numer – volume: 76 start-page: 3 year: 1982 ident: 10.1002/jcd.21425-BIB0018|jcd21425-cit-0018 article-title: The method of quadratic forms in a combinatorial problem of Turán publication-title: Vestnik Moskov. Univ. Ser. I Mat Mekh – volume: 11 start-page: 179 year: 1995 ident: 10.1002/jcd.21425-BIB0019|jcd21425-cit-0019 article-title: What we know and what we do not know about Turán numbers publication-title: Graphs Combin doi: 10.1007/BF01929486 – volume: 9 start-page: 265 year: 2000 ident: 10.1002/jcd.21425-BIB0012|jcd21425-cit-0012 article-title: Explicit constructions of Rödl's asymptotically good packings and coverings publication-title: Combin Probab Comput doi: 10.1017/S0963548300004235 – volume: 3 start-page: 269 year: 1995 ident: 10.1002/jcd.21425-BIB0006|jcd21425-cit-0006 article-title: New constructions for covering designs publication-title: J Combin Des doi: 10.1002/jcd.3180030404 – ident: 10.1002/jcd.21425-BIB0002|jcd21425-cit-0002 doi: 10.1017/CBO9780511586507 |
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| Snippet | A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings... |
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| SubjectTerms | 5E105 Combinatorics covering design Mathematics Turán-system uniform-oriented matroidMathematical Subject Classification: 05B05 uniform‐oriented matroidMathematical Subject Classification: 05B05, 5E105 |
| Title | Connected Covering Numbers |
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