Extended bicolorings of Steiner triple systems of order 2h − 1

Csilla Bujtás, Mario Gionfriddo, Elena Guardo, Lorenzo Milazzo, Z. Tuza, Vitaly Voloshin

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Abstract

A bicoloring of a Steiner triple system STS(n) on n vertices is a coloring of vertices in such a way that every block receives precisely two colors. The maxi-mum (resp. minimum) number of colors in a bicoloring of an STS(n) is denoted by χ̅ (resp. χ). All bicolorable STS(2h − 1)s have upper chromatic number χ̅ ≤ h; also, if χ̅ = h < 10, then lower and upper chromatic numbers coincide, namely, χ = χ̅ = h. In 2003, M. Gionfriddo conjectured that this equality holds whenever χ̅ = h ≥ 2. In this paper we discuss some extensions of bicolorings of STS(v) to bicoloring of STS(2v+1) obtained by using the `doubling plus one construction’. We prove several necessary conditions for bicolorings of STS(2v+1) provided that no new color is used. In addition, for any natural number h we determine a triple system STS(2h+1 − 1) which admits no extended bicolorings.

Original languageEnglish
Pages (from-to)1265-1276
Number of pages12
JournalTaiwanese Journal of Mathematics
Volume21
Issue number6
DOIs
Publication statusPublished - Dec 1 2017

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Keywords

  • Coloring
  • Extended bicoloring
  • Mixed hypergraph
  • Steiner triple system
  • Upper chromatic number

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Bujtás, C., Gionfriddo, M., Guardo, E., Milazzo, L., Tuza, Z., & Voloshin, V. (2017). Extended bicolorings of Steiner triple systems of order 2h − 1. Taiwanese Journal of Mathematics, 21(6), 1265-1276. https://doi.org/10.11650/tjm/8042