The configuration polytope of $\ell$-line configurations in Steiner triple systems

Rok, strany: 2009, 77 - 108

It has been shown that the number of occurrences of any $\ell$-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full $m$-line configurations for $1 ≤ m ≤ \ell$; full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in $v$, the order of the Steiner triple system. Moreover, the counts of full configurations, together with $v$, form a linear basis for all of the configuration counts when $\ell ≤ 7$. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when $\ell = 6$. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored.

ISO 690:

Colbourn, C. 2009. The configuration polytope of $\ell$-line configurations in Steiner triple systems. In Mathematica Slovaca, vol. 59, no.1, pp. 77-108. 0139-9918. DOI: https://doi.org/10.2478/s12175-008-0111-2

APA:

Colbourn, C. (2009). The configuration polytope of $\ell$-line configurations in Steiner triple systems. Mathematica Slovaca, 59(1), 77-108. 0139-9918. DOI: https://doi.org/10.2478/s12175-008-0111-2