Characterization results for Steiner triple systems and their application to edge-colorings of cubic graphs

Daniel Král' (1) , Edita Máčajová (2) , Attila Pór (3) , Jean-Sébastien Sereni (4, 5)

1 ITI - Institut teoretické informatiky
2 Comenius University in Bratislava
3 WKU - Department of Mathematics - Western Kentucky University
4 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
5 KAM - Department of Applied Mathematics (KAM)

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations. Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective non-affine point-transitive Steiner triple system S.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Daniel Král', Edita Máčajová, Attila Pór, Jean-Sébastien Sereni. Characterization results for Steiner triple systems and their application to edge-colorings of cubic graphs. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, 2010, 62 (2), pp.355--381. ⟨10.4153/CJM-2010-021-9⟩. ⟨hal-00487085⟩