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. All bicolorable ST S(2h − 1)s have the upper chromatic number (maximum number of colors) χ ̄ = h, and for h < 10, it was proved that their upper and lower chromatic numbers coincide, i.e. no bicoloring exists with fewer colors, χ ̄ = χ = h. In 2003, M. Gionfriddo conjectured that this equality holds for every integer h ≥ 2. In this paper we discuss some extensions of bicolorings of STS(v) to bicoloring of S T S (2v + 1) obtained by using the ‘doubling plus one construction’. We prove several necessary conditions for bicolorings of S T S (2v + 1) provided that no new color is used. In addition, for any natural number h we determine a triple system ST S(2h+1 − 1) which admits no extended bicolorings.
|Titolo:||Extended bicolorings of Steiner triple systems of order 2^h - 1|
|Data di pubblicazione:||Being printed|
|Appare nelle tipologie:||1.1 Articolo in rivista|