Show Your Colors

The object of this game is to compile a list of different words -- each related to its predecessor to form a compound word or familiar phrase -- and have this chain begin and end with the same word. Our theme is Color ... and, to keep the exercise interesting, we ask that your chains be no fewer than fifteen, nor more than twenty-five words in length. This will eliminate a simple BLUE-sky-BLUE combination ... and it conserves on paper by keeping your lists from stretching to infinity

Tiles and colors

Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles are rectangles and isosceles triangles. Some of these models have been solved recently by means of Bethe Ansatz. We discuss the question why only these models in a larger family are solvable, and we search for the Yang-Baxter structure behind their integrablity. In this quest we find the Bethe Ansatz solution of the problem of coloring the edges of the square lattice in four colors, such that edges of the same color never meet in the same vertex.Comment: 18 pages, 3 figures (in 5 eps files

Algorithms for Coloring Quadtrees

We describe simple linear time algorithms for coloring the squares of balanced and unbalanced quadtrees so that no two adjacent squares are given the same color. If squares sharing sides are defined as adjacent, we color balanced quadtrees with three colors, and unbalanced quadtrees with four colors; these results are both tight, as some quadtrees require this many colors. If squares sharing corners are defined as adjacent, we color balanced or unbalanced quadtrees with six colors; for some quadtrees, at least five colors are required.Comment: 7 pages, 9 figure

The Correlations between the Intrinsic Colors and Spectroscopic Metallicities of M31 Globular Clusters

We present the correlations between the spectroscopic metallicities and ninety-three different intrinsic colors of M31 globular clusters, including seventy-eight BATC colors and fifteen SDSS and near infrared ugrizK colors. The BATC colors were derived from the archival images of thirteen filters (from c to p), which were taken by Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey with a 60/90 cm f/3 Schmidt telescope. The spectroscopic metallicities adopted in our work were from literature. We fitted the correlations of seventy-eight different BATC colors and the metallicities for 123 old confirmed globular clusters, and the result implies that correlation coefficients of twenty-three colors r>0.7. Especially, for the colors $(f-k)_0$, $(f-o)_0$, and $(h-k)_0$, the correlation coefficients are r>0.8. Meanwhile, we also note that the correlation coefficients (r) approach zero for $(g-h)_0$, $(k-m)_0$, $(k-n)_0$, and $(m-n)_0$, which are likely to be independent of metallicity. Similarity, we fitted the correlations of metallicity and ugrizK colors for 127 old confirmed GCs. The result indicates that all these colors are metal-sensitive (r>0.7), of which $(u-K)_0$ is the most metal-sensitive color. Our work provides an easy way to simply estimate the metallicity from colors.Comment: 25 pages, 11 figures, 2 tables, accepted for publication in PASP

Integrated Light 2MASS IR Photometry of Galactic Globular Clusters

We have mosaiced 2MASS images to derive surface brightness profiles in JHK for 104 Galactic globular clusters. We fit these with King profiles, and show that the core radii are identical to within the errors for each of these IR colors, and are identical to the core radii at V in essentially all cases. We derive integrated light colors V-J, V-H, V-K_s, J-H and J-K_s for these globular clusters. Each color shows a reasonably tight relation between the dereddened colors and metallicity. Fits to these are given for each color. The IR--IR colors have very small errors due largely to the all-sky photometric calibration of the 2MASS survey, while the V-IR colors have substantially larger uncertainties. We find fairly good agreement with measurements of integrated light colors for a smaller sample of Galactic globular clusters by Aaronson, Malkan & Kleinmann from 1977. Our results provide a calibration for the integrated light of distant single burst old stellar populations from very low to Solar metallicities. A comparison of our dereddened measured colors with predictions from several models of the integrated light of single burst old populations shows good agreement in the low metallicity domain for V-K_s colors, but an offset at a fixed [Fe/H] of ~0.1 mag in J-K_s, which we ascribe to photometric system transformation issues. Some of the models fail to reproduce the behavior of the integrated light colors of the Galactic globular clusters near Solar metallicity.Comment: Accepted for publication in the A

K3K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a $K_3$-WORM coloring with two colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable graph in which the minimum number of colors is exactly $k$. There also exist $K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains intractable even for graphs of maximum degree 9 if $k=3$). On the other hand, we prove positive results for $d$-degenerate graphs with small $d$, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.Comment: 18 page