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1.
Diane M. Donovan Mike J. Grannell Terry S. Griggs James G. Lefevre Thomas McCourt 《组合设计杂志》2011,19(1):16-27
It is shown that for every admissible order v for which a cyclic Steiner triple system exists, there exists a biembedding of a cyclic Steiner quasigroup of order v with a copy of itself. Furthermore, it is shown that for each n≥2 the projective Steiner quasigroup of order 2n?1 has a biembedding with a copy of itself. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:16‐27, 2010 相似文献
2.
Darryn Bryant 《组合设计杂志》2002,10(5):313-321
A well‐known, and unresolved, conjecture states that every partial Steiner triple system of order u can be embedded in a Steiner triple system of order υ for all υ ≡ 1 or 3, (mod 6), υ ≥ 2u + 1. However, some partial Steiner triple systems of order u can be embedded in Steiner triple systems of order υ <2u + 1. A more general conjecture that considers these small embeddings is presented and verified for some cases. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 313–321, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10017 相似文献
3.
In this article, we show that every simple r‐regular graph G admits a balanced P4‐decomposition if r ≡ 0(mod 3) and G has no cut‐edge when r is odd. We also show that a connected 4‐regular graph G admits a P4‐decomposition if and only if |E(G)| ≡ 0(mod 3) by characterizing graphs of maximum degree 4 that admit a triangle‐free Eulerian tour. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 135–143, 1999 相似文献
4.
A Steiner triple system of order ν, denoted STS(ν), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary
and sufficient conditions for the existence of a tricyclic STS(ν) for several cases. We also pose conjectures concerning their existence in two remaining cases. 相似文献
5.
We prove that there is a Steiner triple system ?? such that every simple cubic graph can have its edges colored by points of ?? in such a way that for each vertex the colors of the three incident edges form a triple in ??. This result complements the result of Holroyd and ?koviera that every bridgeless cubic graph admits a similar coloring by any Steiner triple system of order greater than 3. The Steiner triple system employed in our proof has order 381 and is probably not the smallest possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 15–24, 2004 相似文献
6.
Mike J. Grannell Terry S. Griggs Edita Máčajová Martin Škoviera 《Journal of Graph Theory》2013,74(2):163-181
An ‐coloring of a cubic graph G is an edge coloring of G by points of a Steiner triple system such that the colors of any three edges meeting at a vertex form a block of . A Steiner triple system that colors every simple cubic graph is said to be universal. It is known that every nontrivial point‐transitive Steiner triple system that is neither projective nor affine is universal. In this article, we present the following results.
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7.
Marco Buratti 《组合设计杂志》2001,9(3):215-226
Phelps and Rosa introduced the concept of 1‐rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν ? 1 [Discrete Math. 33 ( 12 ), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1‐rotational STS(ν) in a more general sense. An STS(ν) is 1‐rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1‐rotational over the cyclic group. We denote by ??1r, ??1r, ??1r, ??1r, the spectrum of values of ν for which there exists a 1‐rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine ??1r and find partial answers about ??1r and ??1r. The smallest 1‐rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1‐rotational STS(25) is over SL2(3), the special linear group of dimension 2 over Z3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001 相似文献
8.
The codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2m?1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2m?1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007 相似文献
9.
Peter J. Cameron 《组合设计杂志》2005,13(6):466-470
It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν > g(u). We find exponential upper and lower bounds for g. © 2005 Wiley Periodicals, Inc. J Combin Designs. 相似文献
10.
We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005. 相似文献
11.
M. H. Armanious 《Periodica Mathematica Hungarica》1992,25(3):257-261
In this paper it will be constructed an abstract geometry will be called a triple space, which is defined in general sense by the closure theoretic definition of geometry “see [4]”. And it is proved that the category of triple spaces is isomorphic to the category of Steiner triple systems. And hence it could be shown that the class of Steiner triple systems which satisfy the geometric axiomI 3, (I3) $$\forall x_1 ,x_2 ,x_{3,} y;ify \in< x_1 ,x_2 ,x_3 > \backslash< x_1 ,x_2 > \Rightarrow x_3 \in< x_1 ,x_3 ,y > $$ is exactly the class of all Steiner triple systems in which every triangle generate a planar subsystem. 相似文献
12.
Adam Wolfe 《组合设计杂志》2006,14(3):229-236
This paper gives a proof of the existence of anti‐mitre Steiner triple systems of order n for every n ≡ 1,3 mod 6 except for n = 9. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 229–236, 2006 相似文献
13.
We show that an anti‐Pasch Steiner triple system of order v exists for v ≡ 1 or 3 (mod 6), apart from v = 7 and 13. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 300–309, 2000 相似文献
14.
A Steiner triple system S is a C-ubiquitous (where C is a configuration) if every line of S is contained in a copy of C, and is n-ubiquitous if it is C-ubiquitous for every n-line configuration C. We determine the spectrum of 4-ubiquitous Steiner triple systems as well as the spectra of C-ubiquitous Steiner triple systems for all configurations C with five lines. © 1997 John Wiley & Sons, Inc. 相似文献
15.
Yuichiro Fujiwara 《Journal of Algebraic Combinatorics》2007,26(4):495-506
In 1973 Paul Erdős conjectured that there is an integer v
0(r) such that, for every v>v
0(r) and v≡1,3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i+2 points for every 1<i≤r. Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r≥13 there exists no point-transitive r-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results
on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds
on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown
that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS
from being high-sparse.
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16.
We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D. Stones. 相似文献
17.
An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family, first defined by Bahmanian and ?ajna, is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we study the notions of a spanning Euler tour and a spanning Euler family, that is, an Euler tour (family) that also traverses each vertex of the hypergraph at least once. We examine necessary and sufficient conditions for a hypergraph to admit a spanning Euler family, most notably when the hypergraph possesses a vertex cut consisting of vertices of degree two. Moreover, we characterise hypergraphs with a vertex cut of cardinality at most two that admit a spanning Euler tour (family). This result enables us to reduce the problem of existence of a spanning Euler tour (which is NP-complete), as well as the problem of a spanning Euler family, to smaller hypergraphs. 相似文献
18.
Vaclav Linek 《组合设计杂志》2007,15(5):369-392
A Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X, ) with |X| = v and a set of quadruples taken from X such that every triple in X is in a unique quadruple in . Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v < 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007 相似文献
19.
Frantisek Franek 《组合设计杂志》2000,8(3):157-165
We are interested in the sizes of cliques that are to be found in any arbitrary spanning graph of a Steiner triple system 𝒮. In this paper we investigate spanning graphs of projective Steiner triple systems, proving, not surprisingly, that for any positive integer k and any sufficiently large projective Steiner triple system 𝒮, every spanning graph of 𝒮 contains a clique of size k. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 157–165, 2000 相似文献
20.
A cyclic face 2‐colourable triangulation of the complete graph Kn in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi‐embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi‐embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi‐embedding and we show how all non‐isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001 相似文献