Triangular Graph

The triangular graph with $n$ nodes on a side is denoted $T(n)$. Tutte (1970) showed that the Chromatic Polynomials of planar triangular graphs possess a Root close to $\phi^2=2.618033\ldots$, where $\phi$ is the Golden Mean. More precisely, if $n$ is the number of Vertices of $G$, then

$$P_G(\phi^2)\leq \phi^{5-n}$$

(Le Lionnais 1983, p. 46). Every planar triangular graph possesses a Vertex of degree 3, 4, or 5 (Le Lionnais 1983, pp. 49 and 53).

See also Lattice Graph

References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Tutte, W. T. ``On Chromatic Polynomials and the Golden Ratio.'' J. Combin. Theory 9, 289-296, 1970.