Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family in one of the known finite circle geometries of order , with , must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.
{"title":"Stability of Erdős–Ko–Rado theorems in circle geometries","authors":"Sam Adriaensen","doi":"10.1002/jcd.21854","DOIUrl":"https://doi.org/10.1002/jcd.21854","url":null,"abstract":"<p>Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math> in one of the known finite circle geometries of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>, with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mfrac>\u0000 <mn>1</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 </mfrac>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $| {rm{ {mathcal F} }},| ge frac{1}{sqrt{2}}{q}^{2}+2sqrt{2}q+8$</annotation>\u0000 </semantics></math>, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 11","pages":"689-715"},"PeriodicalIF":0.7,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72149789","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}
It is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group G $G$ of a 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ design D ${mathscr{D}}$ , with λ∣k $lambda | k$ , is either an affine group or an almost simple classical group. Moreover, when G $G$ is the smallest Ree group, D ${mathscr{D}}$ is isomorphic either to the 2‐ (62,6,2) $({6}^{2},6,2)$ design or to one of the three 2‐ (62,6,6) $({6}^{2},6,6)$ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic C ${mathscr{C}}$ of PG2(8) $P{G}_{2}(8)$ as a point set and 6‐sets of secants in a remarkable configuration as a block set.
{"title":"On flag‐transitive 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ designs with λ∣k $lambda | k$","authors":"Alessandro Montinaro, Eliana Francot","doi":"10.1002/jcd.21852","DOIUrl":"https://doi.org/10.1002/jcd.21852","url":null,"abstract":"It is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group G $G$ of a 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ design D ${mathscr{D}}$ , with λ∣k $lambda | k$ , is either an affine group or an almost simple classical group. Moreover, when G $G$ is the smallest Ree group, D ${mathscr{D}}$ is isomorphic either to the 2‐ (62,6,2) $({6}^{2},6,2)$ design or to one of the three 2‐ (62,6,6) $({6}^{2},6,6)$ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic C ${mathscr{C}}$ of PG2(8) $P{G}_{2}(8)$ as a point set and 6‐sets of secants in a remarkable configuration as a block set.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"20 1","pages":"653 - 670"},"PeriodicalIF":0.7,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76703413","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}
<p>It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> of a 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <annotation> $({k}^{2},k,lambda )$</annotation> </semantics></math> design <math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> ${mathscr{D}}$</annotation> </semantics></math>, with <math> <semantics> <mrow> <mi>λ</mi> <mo>∣</mo> <mi>k</mi> </mrow> <annotation> $lambda | k$</annotation> </semantics></math>, is either an affine group or an almost simple classical group. Moreover, when <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> is the smallest Ree group, <math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> ${mathscr{D}}$</annotation> </semantics></math> is isomorphic either to the 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <annotation> $({6}^{2},6,2)$</annotation> </semantics></math> design or to one of the three 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <annotation> $({6}^{2},6,6)$</annotation> </semantics></math> designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic <math> <sem
<p>An idempotent Latin square of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> is called resolvable and denoted by RILS(<i>v</i>) if the <math> <semantics> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $v(v-1)$</annotation> </semantics></math> off-diagonal cells can be resolved into <math> <semantics> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <annotation> $v-1$</annotation> </semantics></math> disjoint transversals. A large set of resolvable idempotent Latin squares of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>, briefly LRILS(<i>v</i>), is a collection of <math> <semantics> <mrow> <mi>v</mi> <mo>−</mo> <mn>2</mn> </mrow> <annotation> $v-2$</annotation> </semantics></math> RILS(<i>v</i>)s pairwise agreeing on only the main diagonal. In this article, an LRILS(<i>v</i>) is constructed for <math> <semantics> <mrow> <mi>v</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>14</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>22</mn> <mo>,</mo> <mn>28</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mn>35</mn> <mo>,</mo> <mn>38</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo>
{"title":"The spectrum for large sets of resolvable idempotent Latin squares","authors":"Xiangqian Li, Yanxun Chang","doi":"10.1002/jcd.21853","DOIUrl":"https://doi.org/10.1002/jcd.21853","url":null,"abstract":"<p>An idempotent Latin square of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> is called resolvable and denoted by RILS(<i>v</i>) if the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $v(v-1)$</annotation>\u0000 </semantics></math> off-diagonal cells can be resolved into <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $v-1$</annotation>\u0000 </semantics></math> disjoint transversals. A large set of resolvable idempotent Latin squares of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>, briefly LRILS(<i>v</i>), is a collection of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $v-2$</annotation>\u0000 </semantics></math> RILS(<i>v</i>)s pairwise agreeing on only the main diagonal. In this article, an LRILS(<i>v</i>) is constructed for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mn>14</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>20</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>22</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>28</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>34</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>35</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>38</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>40</mn>\u0000 \u0000 <mo>,</mo>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 10","pages":"671-683"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166526","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}
An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v − 1 ) $v(v-1)$ off‐diagonal cells can be resolved into v − 1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$ , briefly LRILS(v), is a collection of v − 2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v ∈{14 , 20 , 22 , 28 , 34 , 35 , 38 , 40 , 42 , 46 , 50 , 55 , 62 } $vin {14,20,22,28,34,35,38,40,42,46,50,55,62}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v ≥ 3 $vge 3$ , except v = 6 $v=6$ .
如果v(v-1)$ v(v-1)$离对角线单元可以分解成v-1$ v-1$不相交的截线，则v$ v$阶的幂等拉丁方阵称为可分解的，用RILS(v)表示。一个大的v$ v$阶可解幂等拉丁平方集，简称LRILS(v)，是v−2$ v-2$ RILS(v)的集合，它们只在主对角线上成对一致。本文利用乘子自同构群对v∈{14,20,22,28,34,35,38,40,42,46,50,55,62}$ v In {14,20,22,28,34,35,38,40,42,46,50,55,62}$构造了一个LRILS(v)。因此，除了v=6$ v=6$外，对于任何正整数v≥3$ v ge3 $都存在LRILS(v)。
{"title":"The spectrum for large sets of resolvable idempotent Latin squares","authors":"Xiangqian Li, Yanxun Chang","doi":"10.1002/jcd.21853","DOIUrl":"https://doi.org/10.1002/jcd.21853","url":null,"abstract":"An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v − 1 ) $v(v-1)$ off‐diagonal cells can be resolved into v − 1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$ , briefly LRILS(v), is a collection of v − 2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v ∈{14 , 20 , 22 , 28 , 34 , 35 , 38 , 40 , 42 , 46 , 50 , 55 , 62 } $vin {14,20,22,28,34,35,38,40,42,46,50,55,62}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v ≥ 3 $vge 3$ , except v = 6 $v=6$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"71 1","pages":"671 - 683"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83991533","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 k $k$ ‐extended q $q$ ‐near Skolem sequence of order n $n$ , denoted by Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ , is a sequence s1,s2,…,s2n−1 ${s}_{1},{s}_{2},ldots ,{s}_{2n-1}$ where sk=0 ${s}_{k}=0$ and for each integer ℓ∈[1,n]{q} $ell in [1,n]backslash {q}$ there are two indices i $i$ , j $j$ such that si=sj=ℓ ${s}_{i}={s}_{j}=ell $ and ∣i−j∣=ℓ $| i-j| =ell $ . For an Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ to exist it is necessary that q≡k(mod2) $qequiv k,(mathrm{mod},2)$ when n≡0,1(mod4) $nequiv 0,1,(mathrm{mod},4)$ and q≢k(mod2) $qnotequiv k,(mathrm{mod},2)$ when n≡2,3(mod4) $nequiv 2,3,(mathrm{mod},4)$ , where (n,q,k)≠(3,2,3) $(n,q,k)ne (3,2,3)$ , (4,2,4) $(4,2,4)$ . Any triple (n,q,k) $(n,q,k)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ for all admissible (n,q,k) $(n,q,k)$ with q∈⌊n+23⌋,⌊n−22⌋ $qin left[lfloor frac{n+2}{3}rfloor ,lfloor frac{n-2}{2}rfloor right]$ and k∈⌊2n3⌋,n−1 $kin left[lfloor frac{2n}{3}rfloor ,n-1right]$ .
{"title":"Extended near Skolem sequences, Part III","authors":"C. Baker, V. Linek, N. Shalaby","doi":"10.1002/jcd.21851","DOIUrl":"https://doi.org/10.1002/jcd.21851","url":null,"abstract":"A k $k$ ‐extended q $q$ ‐near Skolem sequence of order n $n$ , denoted by Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ , is a sequence s1,s2,…,s2n−1 ${s}_{1},{s}_{2},ldots ,{s}_{2n-1}$ where sk=0 ${s}_{k}=0$ and for each integer ℓ∈[1,n]{q} $ell in [1,n]backslash {q}$ there are two indices i $i$ , j $j$ such that si=sj=ℓ ${s}_{i}={s}_{j}=ell $ and ∣i−j∣=ℓ $| i-j| =ell $ . For an Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ to exist it is necessary that q≡k(mod2) $qequiv k,(mathrm{mod},2)$ when n≡0,1(mod4) $nequiv 0,1,(mathrm{mod},4)$ and q≢k(mod2) $qnotequiv k,(mathrm{mod},2)$ when n≡2,3(mod4) $nequiv 2,3,(mathrm{mod},4)$ , where (n,q,k)≠(3,2,3) $(n,q,k)ne (3,2,3)$ , (4,2,4) $(4,2,4)$ . Any triple (n,q,k) $(n,q,k)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ for all admissible (n,q,k) $(n,q,k)$ with q∈⌊n+23⌋,⌊n−22⌋ $qin left[lfloor frac{n+2}{3}rfloor ,lfloor frac{n-2}{2}rfloor right]$ and k∈⌊2n3⌋,n−1 $kin left[lfloor frac{2n}{3}rfloor ,n-1right]$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"80 1","pages":"637 - 652"},"PeriodicalIF":0.7,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81213966","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}
An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance and length can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and -ary quasigroups.
{"title":"Embedding in MDS codes and Latin cubes","authors":"Vladimir N. Potapov","doi":"10.1002/jcd.21849","DOIUrl":"https://doi.org/10.1002/jcd.21849","url":null,"abstract":"<p>An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math> and length <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-ary quasigroups.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 9","pages":"626-633"},"PeriodicalIF":0.7,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72159165","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 subset A $A$ of an abelian group G $G$ is sequenceable if there is an ordering ( a 1 , … , a k ) $({a}_{1},ldots ,{a}_{k})$ of its elements such that the partial sums ( s 0 , s 1 , … , s k ) $({s}_{0},{s}_{1},ldots ,{s}_{k})$ , given by s 0 = 0 ${s}_{0}=0$ and s i = ∑ j = 1 i a j ${s}_{i}={sum }_{j=1}^{i}{a}_{j}$ for 1 ≤ i ≤ k $1le ile k$ , are distinct, with the possible exception that we may have s k = s 0 = 0 ${s}_{k}={s}_{0}=0$ . In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized by Alspach and Liversidge into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set A $A$ do not sum to 0 then there exists a simple path P $P$ in the Cayley graph C a y [ G : ± A ] $Cay[G:pm A]$ such that Δ ( P ) = ± A ${rm{Delta }}(P)=pm A$ . In this paper, inspired by this graph–theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk W $W$ of girth bigger than t $t$ (for a given t < k $tlt k$ ) and such that Δ ( W ) = ± A ${rm{Delta }}(W)=pm A$ . This is possible given that the partial sums s i ${s}_{i}$ and s j ${s}_{j}$ are different whenever i $i$ and j $j$ are distinct and ∣ i − j ∣ ≤ t $| i-j| le t$ . In this case, we say that the set A $A$ is t $t$ ‐weakly sequenceable. The main result here presented is that any subset A $A$ of Z p ⧹ { 0 } ${{mathbb{Z}}}_{p}setminus {0}$ is t $t$ ‐weakly sequenceable whenever t < 7 $tlt 7$ or when A $A$ does not contain pairs of type { x , − x } ${x,-x}$ and t < 8 $tlt 8$ .
一个阿贝尔群G $G$的子集A $A$是可序的，如果它的元素有一个序(A 1，…，A k) $({a}_{1},ldots ,{a}_{k})$，使得部分和(s 0, s 1，…，s k) $({s}_{0},{s}_{1},ldots ,{s}_{k})$ (s 0 = 0 ${s}_{0}=0$和si =∑j = 1 i A j ${s}_{i}={sum }_{j=1}^{i}{a}_{j}$对于1≤i≤k $1le ile k$)是不同的，除了可能的例外，我们可以有s k = s 0 = 0 ${s}_{k}={s}_{0}=0$。在文献中，关于阿贝尔群子集的可序列性有几个猜想和问题，Alspach和Liversidge将这些猜想组合并总结为“如果一个阿贝尔群的子集不包含0，那么它是可序列的”的猜想。如果一个可序列集合a $A$的元素和不等于0，那么在Cayley图Cay [G:±a] $Cay[G:pm A]$中存在一条简单路径P $P$，使得Δ (P) =±a ${rm{Delta }}(P)=pm A$。在本文中，受这个图论解释的启发，我们提出了一个弱化这个猜想的方法。这里，在上述假设下，我们想要找到一个排序，它的部分和定义了一个周长大于t $t$的行走W $W$(对于给定的t < k $tlt k$)，并且使得Δ (W) =±a ${rm{Delta }}(W)=pm A$。这是可能的，当i $i$和j $j$不同且∣i−j∣≤t $| i-j| le t$时，部分和si ${s}_{i}$和s j ${s}_{j}$是不同的。在这种情况下，我们说集合A $A$是t $t$‐弱可测序的。这里给出的主要结果是，当t < 7 $tlt 7$或当A {}$A$不包含类型x， - x${x,-x}$和t < 8 $tlt 8$的对时，Z p⧹0${{mathbb{Z}}}_{p}setminus {0}$的任何子集A {}$A$都是t $t$‐弱可测序的。
{"title":"Weak sequenceability in cyclic groups","authors":"Simone Costa, Stefano Della Fiore","doi":"10.1002/jcd.21862","DOIUrl":"https://doi.org/10.1002/jcd.21862","url":null,"abstract":"A subset A $A$ of an abelian group G $G$ is sequenceable if there is an ordering ( a 1 , … , a k ) $({a}_{1},ldots ,{a}_{k})$ of its elements such that the partial sums ( s 0 , s 1 , … , s k ) $({s}_{0},{s}_{1},ldots ,{s}_{k})$ , given by s 0 = 0 ${s}_{0}=0$ and s i = ∑ j = 1 i a j ${s}_{i}={sum }_{j=1}^{i}{a}_{j}$ for 1 ≤ i ≤ k $1le ile k$ , are distinct, with the possible exception that we may have s k = s 0 = 0 ${s}_{k}={s}_{0}=0$ . In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized by Alspach and Liversidge into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set A $A$ do not sum to 0 then there exists a simple path P $P$ in the Cayley graph C a y [ G : ± A ] $Cay[G:pm A]$ such that Δ ( P ) = ± A ${rm{Delta }}(P)=pm A$ . In this paper, inspired by this graph–theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk W $W$ of girth bigger than t $t$ (for a given t < k $tlt k$ ) and such that Δ ( W ) = ± A ${rm{Delta }}(W)=pm A$ . This is possible given that the partial sums s i ${s}_{i}$ and s j ${s}_{j}$ are different whenever i $i$ and j $j$ are distinct and ∣ i − j ∣ ≤ t $| i-j| le t$ . In this case, we say that the set A $A$ is t $t$ ‐weakly sequenceable. The main result here presented is that any subset A $A$ of Z p ⧹ { 0 } ${{mathbb{Z}}}_{p}setminus {0}$ is t $t$ ‐weakly sequenceable whenever t < 7 $tlt 7$ or when A $A$ does not contain pairs of type { x , − x } ${x,-x}$ and t < 8 $tlt 8$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"4 1","pages":"735 - 751"},"PeriodicalIF":0.7,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87428608","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}
<p>In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> (SQS<math> <semantics> <mrow> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $(v)$</annotation> </semantics></math>) for all <math> <semantics> <mrow> <mi>v</mi> <mo>≡</mo> <mn>4</mn> </mrow> <annotation> $vequiv 4$</annotation> </semantics></math> or <math> <semantics> <mrow> <mn>8</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mn>12</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $8,(mathrm{mod},12)$</annotation> </semantics></math>. The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and <math> <semantics> <mrow> <mi>s</mi> </mrow> <annotation> $s$</annotation> </semantics></math>-fan designs. In this paper, it is proved that a 2-chromatic SQS<math> <semantics> <mrow> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $(v)$</annotation> </semantics></math> also exists if <math> <semantics> <mrow> <mrow> <mi>v</mi> <mo>≡</mo> <mn>10</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mn>12</mn> </mrow>
{"title":"New infinite classes of 2-chromatic Steiner quadruple systems","authors":"Lijun Ji, Shuangqing Liu, Ye Yang","doi":"10.1002/jcd.21845","DOIUrl":"https://doi.org/10.1002/jcd.21845","url":null,"abstract":"<p>In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> (SQS<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v)$</annotation>\u0000 </semantics></math>) for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $vequiv 4$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>8</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>12</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $8,(mathrm{mod},12)$</annotation>\u0000 </semantics></math>. The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math>-fan designs. In this paper, it is proved that a 2-chromatic SQS<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v)$</annotation>\u0000 </semantics></math> also exists if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>10</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>12</mn>\u0000 </mrow>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 9","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":"72190472","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}
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