Coloring Properties of Mixed Cycloids

György Dósa; Nicholas Newman; Zsolt Tuza; Vitaly Voloshin

doi:10.3390/sym13081539

Coloring Properties of Mixed Cycloids

Symmetry ◽

10.3390/sym13081539 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1539

Author(s):

György Dósa ◽

Nicholas Newman ◽

Zsolt Tuza ◽

Vitaly Voloshin

Keyword(s):

Upper Bound ◽

Cyclic Order ◽

Discrete Structures ◽

Mixed Hypergraph

In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any p≥3 and any q≥2. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.

Download Full-text

A least upper bound for the number of singular points on normal arcs and curves of cyclic order four

Geometriae Dedicata ◽

10.1007/bf00181349 ◽

1978 ◽

Vol 7 (1) ◽

Cited By ~ 3

Author(s):

G. Spoar

Keyword(s):

Upper Bound ◽

Singular Points ◽

Cyclic Order ◽

Order Four

Download Full-text

A new upper bound for sorting permutations with prefix transpositions

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500779 ◽

2020 ◽

Vol 12 (06) ◽

pp. 2050077

Author(s):

Pramod P. Nair ◽

Rajan Sundaravaradhan ◽

Bhadrachalam Chitturi

Keyword(s):

Symmetric Group ◽

Upper Bound ◽

A Genome ◽

Minimum Number ◽

Discrete Structures ◽

The Given

Permutations are discrete structures that naturally model a genome where every gene occurs exactly once. In a permutation over the given alphabet [Formula: see text], each symbol of [Formula: see text] appears exactly once. A transposition operation on a given permutation [Formula: see text] exchanges two adjacent sublists of [Formula: see text]. If one of these sublists is restricted to be a prefix then one obtains a prefix transposition. The symmetric group of permutations with [Formula: see text] symbols derived from the alphabet [Formula: see text] is denoted by [Formula: see text]. The symmetric prefix transposition distance between [Formula: see text] and [Formula: see text] is the minimum number of prefix transpositions that are needed to transform [Formula: see text] into [Formula: see text]. It is known that transforming an arbitrary [Formula: see text] into an arbitrary [Formula: see text] is equivalent to sorting some [Formula: see text]. Thus, upper bound for transforming any [Formula: see text] into any [Formula: see text] with prefix transpositions is simply the upper bound to sort any permutation [Formula: see text]. The current upper bound is [Formula: see text] for prefix transposition distance over [Formula: see text]. In this paper, we improve the same to [Formula: see text].

Download Full-text

The High Resolution Appearance of Biological Substrates

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100063998 ◽

1969 ◽

Vol 27 ◽

pp. 416-417

Author(s):

Glen B. Haydon

Keyword(s):

High Resolution ◽

Phase Image ◽

Biological Structure ◽

Electron Microscopic ◽

Thin Sections ◽

Resolution Electron ◽

Discrete Structures ◽

Large Phase ◽

Biological Substrates ◽

High Resolution Electron Micrographs

High resolution electron microscopic study of negatively stained macromolecules and thin sections of tissue embedded in a variety of media are difficult to interpret because of the superimposed phase image granularity. Although all of the information concerning the biological structure of interest may be present in a defocused electron micrograph, the high contrast of large phase image granules produced by the substrate makes it impossible to distinguish the phase ‘points’ from discrete structures of the same dimensions. Theory predicts the findings; however, it does not allow an appreciation of the actual appearance of the image under various conditions. Therefore, though perhaps trivial, training of the cheapest computer produced by mass labor has been undertaken in order to learn to appreciate the factors which affect the appearance of the background in high resolution electron micrographs.

Download Full-text