Graph decompositions in projective geometries

被引:10
作者
Buratti, Marco [1 ]
Nakic, Anamari [2 ]
Wassermann, Alfred [3 ]
机构
[1] Univ Perugia, Dipartimento Matemat & Informat, Via Vanvitelli 1, I-06123 Perugia, Italy
[2] Univ Zagreb, Fac Elect Engn & Comp, Zagreb, Croatia
[3] Univ Bayreuth, Dept Math, Bayreuth, Germany
关键词
design over a finite field; difference family; difference set; graph decomposition; group divisible design over a finite field; projective space; spread; AUTOMORPHISM GROUP; DESIGNS; PARTITIONS; ANALOG;
D O I
10.1002/jcd.21761
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let PG(Fqv) be the (v-1)-dimensional projective space over Fq and let Gamma be a simple graph of order qk-1q-1 for some k. A 2-(v,Gamma,lambda) design over Fq is a collection beta of graphs (blocks) isomorphic to Gamma with the following properties: the vertex set of every block is a subspace of PG(Fqv); every two distinct points of PG(Fqv) are adjacent in exactly lambda blocks. This new definition covers, in particular, the well-known concept of a 2-(v,k,lambda) design over Fq corresponding to the case that Gamma is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of Gamma-decompositions over F2 or F3 for which Gamma is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that Gamma is complete and lambda=1, that is, the Steiner 2-designs over a finite field. Also, we briefly touch the new topic of near resolvable 2-(v,2,1) designs over Fq. This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Gamma-decompositions over a finite field that can be obtained by suitably labeling the vertices of Gamma with the elements of a Singer difference set.
引用
收藏
页码:141 / 174
页数:34
相关论文
共 37 条
  • [1] Abel RJR., 2006, CRC HDB COMBINATORIA, P392
  • [2] PARTITIONING PLANES OF AG2M(2) INTO 2-DESIGNS
    BAKER, RD
    [J]. DISCRETE MATHEMATICS, 1976, 15 (03) : 205 - 211
  • [3] Beth T., 1999, Encyclopedia of Mathematics and Its Applications, V69
  • [4] Systematic construction of q-analogs of t-(v,k,λ)-designs
    Braun, M
    Kerber, A
    Laue, R
    [J]. DESIGNS CODES AND CRYPTOGRAPHY, 2005, 34 (01) : 55 - 70
  • [5] EXISTENCE OF q-ANALOGS OF STEINER SYSTEMS
    Braun, Michael
    Etzion, Tuvi
    Oestergard, Patric R. J.
    Vardy, Alexander
    Wassermann, Alfred
    [J]. FORUM OF MATHEMATICS PI, 2016, 4
  • [6] On the automorphism group of a binary q-analog of the Fano plane
    Braun, Michael
    Kiermaier, Michael
    Nakic, Anamari
    [J]. EUROPEAN JOURNAL OF COMBINATORICS, 2016, 51 : 443 - 457
  • [7] Bryant D., 2006, HDB COMBINATORIAL DE, P477
  • [8] Buratti M, 1998, J COMB DES, V6, P165, DOI 10.1002/(SICI)1520-6610(1998)6:3<165::AID-JCD1>3.0.CO
  • [9] 2-D
  • [10] Buratti M., 2013, ELECT NOTES DISCRE C, V40C, P245