Computing at Glasgow University
Paper ID: 8484
DCS Tech Report Number: TR-2007-235

Triangle-free Linear Spaces on 13 points with line sizes 2,3, and 4

Publication Type: Tech Report (internal)
Appeared in: DCS Technical Report Series
Page Numbers : 1-12
Publisher: Dept of Computing Science, University of Glasgow
Year: 2007

In this document we consider linear spaces with lines of size 2, 3 and 4 for which there is no triangular set of pairs of points {(a,b), (a,c),(b,c)} in the lines of size 2 and 3 which are not all contained in a single line of size 3. We call these structures triangle-free linear spaces with lines of size 2, 3 and 4 ({2,3,4}-TFLSs). We show that the only {2,3,4}-TFLSs on 13 points are the 13-4-1 design and a unique structure consisting of 10 lines of size 4 and 4 of size 3 (and six lines of size 2). We show how, in the latter case, the lines can be manipulated to produce a 13-4-1 design.

Keywords: linear spaces, design, graph,stinson's algorithm

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