Block-Graceful Designs
In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V,B with V=v and B=b as block-graceful if there exists a bijection f:B⟶1,2,…,b such that the induced mapping f+:V⟶Zv given by f+x=∑x∈AA∈BfAmod v is a bijection. A quick observation show...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/4959576 |
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| _version_ | 1832547994724990976 |
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| author | Dilara Erdemir Emre Kolotoğlu |
| author_facet | Dilara Erdemir Emre Kolotoğlu |
| author_sort | Dilara Erdemir |
| collection | DOAJ |
| description | In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V,B with V=v and B=b as block-graceful if there exists a bijection f:B⟶1,2,…,b such that the induced mapping f+:V⟶Zv given by f+x=∑x∈AA∈BfAmod v is a bijection. A quick observation shows that every v,b,r,k,λ−BIBD that is generated from a cyclic difference family is block-graceful when v,r=1. As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v≡1mod 6 and block-graceful projective geometries, i.e., qd+1−1/q−1,qd−1/q−1,qd−1−1/q−1−BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v,k,λ−BIBDs. We consider the case v≡3mod 6 for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order v for v≡3mod 6. We also consider affine geometries and prove that for every integer d≥2 and q≥3, where q is an odd prime power or q=4, there exists a block-graceful qd,q,1−BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures. |
| format | Article |
| id | doaj-art-7a830164595746ee88caed7551e446c3 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-7a830164595746ee88caed7551e446c32025-02-03T06:42:44ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/4959576Block-Graceful DesignsDilara Erdemir0Emre Kolotoğlu1Department of MathematicsDepartment of MathematicsIn this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V,B with V=v and B=b as block-graceful if there exists a bijection f:B⟶1,2,…,b such that the induced mapping f+:V⟶Zv given by f+x=∑x∈AA∈BfAmod v is a bijection. A quick observation shows that every v,b,r,k,λ−BIBD that is generated from a cyclic difference family is block-graceful when v,r=1. As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v≡1mod 6 and block-graceful projective geometries, i.e., qd+1−1/q−1,qd−1/q−1,qd−1−1/q−1−BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v,k,λ−BIBDs. We consider the case v≡3mod 6 for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order v for v≡3mod 6. We also consider affine geometries and prove that for every integer d≥2 and q≥3, where q is an odd prime power or q=4, there exists a block-graceful qd,q,1−BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures.http://dx.doi.org/10.1155/2023/4959576 |
| spellingShingle | Dilara Erdemir Emre Kolotoğlu Block-Graceful Designs Journal of Mathematics |
| title | Block-Graceful Designs |
| title_full | Block-Graceful Designs |
| title_fullStr | Block-Graceful Designs |
| title_full_unstemmed | Block-Graceful Designs |
| title_short | Block-Graceful Designs |
| title_sort | block graceful designs |
| url | http://dx.doi.org/10.1155/2023/4959576 |
| work_keys_str_mv | AT dilaraerdemir blockgracefuldesigns AT emrekolotoglu blockgracefuldesigns |