Open Access
Translator Disclaimer
December, 2017 Extended Bicolorings of Steiner Triple Systems of Order $2^{h}-1$
Csilla Bujtás, Mario Gionfriddo, Elena Guardo, Lorenzo Milazzo, Zsolt Tuza, Vitaly Voloshin
Taiwanese J. Math. 21(6): 1265-1276 (December, 2017). DOI: 10.11650/tjm/8042


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.


Download Citation

Csilla Bujtás. Mario Gionfriddo. Elena Guardo. Lorenzo Milazzo. Zsolt Tuza. Vitaly Voloshin. "Extended Bicolorings of Steiner Triple Systems of Order $2^{h}-1$." Taiwanese J. Math. 21 (6) 1265 - 1276, December, 2017.


Received: 14 May 2016; Revised: 14 February 2017; Accepted: 26 March 2017; Published: December, 2017
First available in Project Euclid: 17 August 2017

zbMATH: 06871368
MathSciNet: MR3732905
Digital Object Identifier: 10.11650/tjm/8042

Primary: 05B07 , 05C15 , 51E10

Keywords: coloring , extended bicoloring , mixed hypergraph , Steiner triple system , upper chromatic number

Rights: Copyright © 2017 The Mathematical Society of the Republic of China


Vol.21 • No. 6 • December, 2017
Back to Top