Appendix of Extending Bicoloring for Steiner Triple Systems

TL;DR

This paper explores extended bicolorings of Steiner triple systems, demonstrating how to generate larger systems with specific coloring properties, leading to new classes of systems with varying chromatic numbers.

Contribution

It introduces the concept of extended bicolorings for STS, providing methods to construct infinite classes of systems with different chromatic number bounds.

Findings

Established existence of extended bicolorings for certain STS

Constructed infinite classes of STS with varying chromatic numbers

Demonstrated the doubling construction preserves coloring properties

Abstract

We initiate the study of extended bicolorings of Steiner triple systems (STS) which start with a $k$-bicoloring of an STS($v$) and end up with a $k$-bicoloring of an STS($2v+1$) obtained by a doubling construction, using only the original colors used in coloring the subsystem STS($v$). By producing many such extended bicolorings, we obtain several infinite classes of orders for which there exist STSs with different lower and upper chromatic number.