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Statement

Theorem 4 (p. 9). "There are 2-fold colouring of $G_{[1,1]}$ with 12 colours and 3-fold colouring of $G_{[1,1]}$ with 16 colours."

Here $G_{[1,1]}$ is the unit distance graph of the plane and a $j$-fold colouring is as in Definition 2 (p. 4): each point receives a $j$-element set of colours, with disjoint sets at distance 1. The ratios are $12/2=6$ and $16/3\approx5.33$ (Table 3, p. 15).

Source. J. Grytczuk, K. Junosza-Szaniawski, J. Sokół, K. Węsek, Fractional and $j$-fold coloring of the plane, Discrete Comput. Geom. 55 (2016), 594-609, doi:10.1007/s00454-016-9769-3; read in arXiv:1506.01887v2 (5 October 2015), Theorem 4 on p. 9 and its proof on pp. 9-10 of that version. The source card records the edition.

Read depth. Claims checked: the statement was read on the print. The proof was read for structure only; its distance claims were not checked.

Proof pointer

pp. 9-10. Both colourings use the tiling of the plane by hexagons of side $1/2$, each hexagon taking its interior, its right border and three of its vertices. For the 2-fold colouring, rows of hexagons receive three colours each, cycling through $\{1,\ldots,12\}$, and a second copy of this layer is shifted by $[3\sqrt3/4,-3/2]$. For the 3-fold colouring, rows receive four colours each from $\{1,\ldots,16\}$, and the paper obtains the second and third layers by moving the coloured grid by $[\sqrt3,-1]$. The paper then checks that hexagons of one colour are far enough apart (Figures 5 and 6).

Dependencies

None.

Bears on

• Problem 508: the problem asks for the chromatic number of $G_{[1,1]}$. These colourings bound the 2-fold and 3-fold chromatic numbers of $G_{[1,1]}$ from above, and so its fractional chromatic number by $16/3$; they give no bound on the chromatic number itself.