In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2‐chromatic Steiner quadruple system of order v $v$ (SQS ( v ) $(v)$ ) for all v ≡ 4 $vequiv 4$ or 8 ( mod 12 ) $8,(mathrm{mod},12)$ . The first author presented a construction for 2‐chromatic SQSs based on 2‐chromatic candelabra quadruple systems and s $s$ ‐fan designs. In this paper, it is proved that a 2‐chromatic SQS ( v ) $(v)$ also exists if v ≡ 10 ( mod 12 ) $vequiv 10,(mathrm{mod},12)$ , or if v ≡ 2 ( mod 24 ) $vequiv 2,(mathrm{mod},24)$ .
{"title":"New infinite classes of 2‐chromatic Steiner quadruple systems","authors":"L. Ji, Shuangqing Liu, Ye Yang","doi":"10.1002/jcd.21845","DOIUrl":"https://doi.org/10.1002/jcd.21845","url":null,"abstract":"In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2‐chromatic Steiner quadruple system of order v $v$ (SQS ( v ) $(v)$ ) for all v ≡ 4 $vequiv 4$ or 8 ( mod 12 ) $8,(mathrm{mod},12)$ . The first author presented a construction for 2‐chromatic SQSs based on 2‐chromatic candelabra quadruple systems and s $s$ ‐fan designs. In this paper, it is proved that a 2‐chromatic SQS ( v ) $(v)$ also exists if v ≡ 10 ( mod 12 ) $vequiv 10,(mathrm{mod},12)$ , or if v ≡ 2 ( mod 24 ) $vequiv 2,(mathrm{mod},24)$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"9 1","pages":"613 - 620"},"PeriodicalIF":0.7,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84180073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Kirkman triple system of order v , KTS (v) , is a resolvable Steiner triple system on v elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS (v) which contain as a subdesign a Steiner triple system of order u , an STS (u) . We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case v= 2u+ 1 , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case v= 2u+ 3 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form v= 2u+w and also near v= 3u . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.
{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"P. Dukes, E. Lamken","doi":"10.1002/jcd.21844","DOIUrl":"https://doi.org/10.1002/jcd.21844","url":null,"abstract":"A Kirkman triple system of order v , KTS (v) , is a resolvable Steiner triple system on v elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS (v) which contain as a subdesign a Steiner triple system of order u , an STS (u) . We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case v= 2u+ 1 , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case v= 2u+ 3 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form v= 2u+w and also near v= 3u . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 1","pages":"581 - 608"},"PeriodicalIF":0.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88790045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.
{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"Peter J. Dukes, Esther R. Lamken","doi":"10.1002/jcd.21844","DOIUrl":"https://doi.org/10.1002/jcd.21844","url":null,"abstract":"<p>A Kirkman triple system of order <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow></math>, KTS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow></math>, is a resolvable Steiner triple system on <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow></math> elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow></math> which contain as a subdesign a Steiner triple system of order <math>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 </mrow></math>, an STS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 </mrow></math>. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow></math> to (at present) three possible exceptions. In addition, we obtain results for other cases of the form <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 <mi>w</mi>\u0000 </mrow></math> and also near <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 <mi>u</mi>\u0000 </mrow></math>. Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"581-608"},"PeriodicalIF":0.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72179676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.
{"title":"Algorithms and complexity for counting configurations in Steiner triple systems","authors":"Daniel Heinlein, Patric R. J. Östergård","doi":"10.1002/jcd.21839","DOIUrl":"https://doi.org/10.1002/jcd.21839","url":null,"abstract":"<p>Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 7","pages":"527-546"},"PeriodicalIF":0.7,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21839","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72156027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order ≤ 8 was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.
{"title":"Classification of minimal blocking sets in small Desarguesian projective planes","authors":"K. Coolsaet, Arne Botteldoorn, V. Fack","doi":"10.1002/jcd.21842","DOIUrl":"https://doi.org/10.1002/jcd.21842","url":null,"abstract":"A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order ≤ 8 was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"106 1","pages":"561 - 580"},"PeriodicalIF":0.7,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72870562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.
{"title":"Classification of minimal blocking sets in small Desarguesian projective planes","authors":"Kris Coolsaet, Arne Botteldoorn, Veerle Fack","doi":"10.1002/jcd.21842","DOIUrl":"https://doi.org/10.1002/jcd.21842","url":null,"abstract":"<p>A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order <math>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>≤</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 </mrow></math> was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"561-580"},"PeriodicalIF":0.7,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72170554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Araujo-Pardo, R. Jajcay, Alejandra Ramos-Rivera, T. Szonyi
A bipartite biregular (m,n;g) $(m,n;g)$ ‐graph Γ ${rm{Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} ${m,n}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$ ‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$ ‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$ ‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$ ‐regular Moore graphs of even girths g=6,8 $g=6,8$ , and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$ ‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $left(k,frac{v-1}{k-1};6right)$ ‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$ ‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$ ‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$ ‐bipartite biregular cages for all integers n≥4 $nge 4$ . Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$ ‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$ ‐, (n,n+2;8) $(n,n+2;8)$ ‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$ ‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$ ‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.
{"title":"On a relation between bipartite biregular cages, block designs and generalized polygons","authors":"G. Araujo-Pardo, R. Jajcay, Alejandra Ramos-Rivera, T. Szonyi","doi":"10.1002/jcd.21836","DOIUrl":"https://doi.org/10.1002/jcd.21836","url":null,"abstract":"A bipartite biregular (m,n;g) $(m,n;g)$ ‐graph Γ ${rm{Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} ${m,n}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$ ‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$ ‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$ ‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$ ‐regular Moore graphs of even girths g=6,8 $g=6,8$ , and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$ ‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $left(k,frac{v-1}{k-1};6right)$ ‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$ ‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$ ‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$ ‐bipartite biregular cages for all integers n≥4 $nge 4$ . Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$ ‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$ ‐, (n,n+2;8) $(n,n+2;8)$ ‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$ ‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$ ‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"6 1","pages":"479 - 496"},"PeriodicalIF":0.7,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81848714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.
{"title":"Steiner triple systems and spreading sets in projective spaces","authors":"Zoltán Lóránt Nagy, Levente Szemerédi","doi":"10.1002/jcd.21841","DOIUrl":"https://doi.org/10.1002/jcd.21841","url":null,"abstract":"<p>We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"549-560"},"PeriodicalIF":0.7,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21841","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72162623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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