Färbbarkeit

Färbbarkeit, or colorability, is a term used in several disciplines to describe the capacity to acquire color. In textiles and materials science, it refers to the ability of a material, especially a fiber or fabric, to absorb, hold, and display color from dyes or pigments. In mathematics and graph theory, it describes the possibility of assigning colors to the elements of a structure under given constraints, typically ensuring that adjacent elements have different colors.

In textile dyeing, dyeability depends on the chemical composition, molecular structure, and surface properties of the substrate. Natural fibers such as cotton dye readily with direct and reactive dyes, while protein fibers like wool and silk respond to acid dyes. Synthetic fibers such as polyester require disperse dyes and often higher temperatures or surface modification. The degree of dye uptake is influenced by crystallinity, dye affinity, pH, temperature, and mordanting. Dyeability is assessed by color strength (K/S), shade uniformity, and washfastness, with pretreatments and dye classes chosen to optimize color yield and repeatability.

In graph theory, a graph is k-colorable if its vertices can be colored with k colors so that adjacent vertices differ. The chromatic number is the smallest such k. Two-colorable graphs are precisely bipartite graphs. Determining colorability for general graphs is NP-complete when k is three or more. The Four Color Theorem states that planar graphs are 4-colorable. Colorability informs practical problems in scheduling, map coloring, frequency assignment, and register allocation.