Extended Bicolorings of Steiner Triple Systems of Order $2^{h}-1$

Abstract

A bicoloring of a Steiner triple system $\operatorname{STS}(n)$ on $n$ vertices is a coloring of vertices in such a way that every block receives precisely two colors. The maximum (resp. minimum) number of colors in a bicoloring of an $\operatorname{STS}(n)$ is denoted by $\overline{\chi}$ (resp. $\chi$). All bicolorable $\operatorname{STS}(2^h-1)$s have upper chromatic number $\overline{\chi} \leq h$; also, if $\overline{\chi} = h \lt 10$, then lower and upper chromatic numbers coincide, namely, $\chi = \overline{\chi} = h$. In 2003, M. Gionfriddo conjectured that this equality holds whenever $\overline{\chi} = h \geq 2$.

In this paper we discuss some extensions of bicolorings of $\operatorname{STS}(v)$ to bicoloring of $\operatorname{STS}(2v+1)$ obtained by using the ‘doubling plus one construction’. We prove several necessary conditions for bicolorings of $\operatorname{STS}(2v+1)$ provided that no new color is used. In addition, for any natural number $h$ we determine a triple system $\operatorname{STS}(2^{h+1}-1$) which admits no extended bicolorings.