A note on "Folding wheels and fans."

In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

On retracts, absolute retracts, and folds in cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic

Perceived Texture Segregation in Chromatic Element-Arrangement Patterns: High Intensity Interference

An element-arrangement pattern is composed of two types of elements that differ in the ways in which they are arranged in different regions of the pattern. We report experiments on the perceived segregation of chromatic element-arrangement patterns composed of equal-size red and blue squares as the luminances of the surround, the interspaces, and the background (surround plus interspaces) are varied. Perceived segregation was markedly reduced by increasing the luminance of the interspaces. Unlike achromatic element-arrangement patterns composed of squares differing in lightness (Beck, Graham, & Sutter, 1991), perceived segregation did not decrease when the luminance of the interspaces was below that of the squares. Perceived segregation was approximately constant for constant ratios of interspace luminance to square luminance and increased with the contrast ratio of the squares. Perceived segregation based on edge alignment was not interfered with by high intensity interspaces. Stereoscopic cues that caused the squares composing the element arrangement pattern to be seen in front of the interspaces did not greatly improve perceived segregation. One explanation of the results is in terms of inhibitory interactions among achromatic and chromatic cortical cells tuned to spatial-frequency and orientation. Alternately, the results may be explained in terms of how the luminance of the interspaces affects the grouping of the squares for encoding surface representations. Neither explanation accounts fully for the data and both mechanisms may be involved.Air Force Office of Scientific Research (F49620-92-J-0334); Northeast Consortium for Engineering Education (A303-21-93); Office of Naval Research (N00014-91J-4100); CNPQ and NUTES/UFRJ, Brazi

Schmidt number of pure bi-partite entangled states and methods of its calculation

An entanglement measure for pure-state continuous-variable bi-partite problem, the Schmidt number, is analytically calculated for one simple model of atom-field scattering.Comment: 3 pages, 1 figure; based on the poster presentation reported on the 11th International Conference on Quantum Optics (ICQO'2006, Minsk, May 26 -- 31, 2006), to be published in special issue of Optics and Spectroscop

Bounds on the achromatic number of partial triple systems

A complete $k$-colouring of a hypergraph is an assignment of $k$ colours to the points such that (1) there is no monochromatic hyperedge, and (2) identifying any two colours produces a monochromatic hyperedge. The achromatic number of a hypergraph is the maximum $k$ such that it admits a complete $k$-colouring. We determine the maximum possible achromatic number among all maximal partial triple systems, give bounds on the maximum and minimum achromatic numbers of Steiner triple systems, and present a possible connection between optimal complete colourings and projective dimension