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Graphs and Combinatorics

, Volume 35, Issue 2, pp 471–478 | Cite as

On the Existence of d-Homogeneous \(\mu \)-Way (v, 3, 2) Steiner Trades

  • S. Golalizadeh
  • N. SoltankhahEmail author
Original Paper
  • 7 Downloads

Abstract

A \(\mu \)-way (vkt) trade is a pair \(T=(X,\{T_1,T_2,\ldots , T_{\mu }\})\) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each \(T_i\)\((1\le i\le \mu )\). In the other words for each \(1\le i<j\le \mu \), the pair \((X,\{T_i,T_j\})\) is a (vkt) trade. A \(\mu \)-way (vkt) trade \(T=(X,\{T_1,T_2,\ldots , T_{\mu }\})\) with any t-subset occuring at most once in \(T_i\)\((1\le i\le \mu )\) is said to be a \(\mu \)-way (vkt) Steiner trade. The trade is called d-homogeneous if each point occurs in exactly d blocks of \(T_i\). In this paper, we construct d-homogeneous \(\mu \)-way (v, 3, 2) Steiner trades with the first, second and third smallest volume for each \(d\equiv 0\) (mod 3) and possible \(\mu \). Also, we show that for each \(d\equiv 0\) (mod 3) there exist d-homogeneous \(\mu \)-way (v, 3, 2) Steiner trades for sufficiently large values of v.

Keywords

Trade Steiner trade d-Homogeneous trade Steiner triple system 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesAlzahra UniversityTehranIran

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