Square Integer Heffter Arrays with Empty Cells

A Heffter array $H(m,n;s,t)$ is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2ms+1}$ such that $i)$ each row contains $s$ filled cells and each column contains $t$ filled cells, $ii)$ every row and column sum to 0, and $iii)$ no element from $\{x,-x\}$ appears twice. Heffter arrays are useful in embedding the complete graph $K_{2nm+1}$ on an orientable surface where the embedding has the property that each edge borders exactly one $s-$cycle and one $t-$cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when $s=m$, i.e. every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is $0$ in $\mathbb{Z}$. We solve most of the instances of this case.Comment: 20 pages, including 2 figure

On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

Vertex-Coloring with Star-Defects

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Shrinkage of Rio Grande silvery minnow after preservation in formalin and storage in ethanol

Understanding how the preservation process affects fish morphology is important for studies that use museum collections as voucher specimens. Sixty-nine Rio Grande silvery minnows (standard length [SL] 32.14–81.65 mm) were observed over 545 days during a formalin-to-water-to-ethanol preservation procedure. Median standard length decreased by the end of each preservation step; specimens shrank 1.06 mm in formalin solution, 1.97 mm in 35% ethanol, 2.17 mm in 50% ethanol, and 2.48 mm in 70% ethanol. Peak shrinkage occurred at 365 d, with a median decrease of 3.86 mm (SL 29.57–75.98 mm). After 545 days, Rio Grande silvery minnows began to increase in length, exhibiting a median shrinkage of 2.04 mm from live length. Research on museum specimens that includes morphological measurements should consider that changes in length or body shape may influence or hinder the ability to detect changes in morphology over time.Se observaron 69 especímenes de carpa chamizal (de longitud estándar entre 32.14–81.65 mm) durante el procedimiento de conservación de formalina a agua y etanol que duró 545 días. La longitud estándar media disminuyó al final de cada paso de conservación; la solución de formalina encogió las muestras 1.06 mm, el etanol al 35% las encogió 1.97 mm, el etanol al 50% las encogió 2.17 mm y el etanol al 70% las encogió 2.48 mm. La disminución máxima se produjo a los 365 días con una reducción media de tamaño de 3.86 mm (de longitud estándar entre 29.57–75.98 mm). Después de 545 días, los especímenes de carpa chamizal comenzaron a aumentar en longitud con una reducción media de tamaño de 2.04 mm de la longitud original. Las investigaciones sobre especímenes de museos que incluyan medidas morfológicas deberán considerar que los cambios en la longitud o en la forma del cuerpo podrían influir o dificultar la detección de cambios morfológicos a lo largo del tiempo

Martin Gardner's minimum no-3-in-a-line problem

In Martin Gardner's October, 1976 Mathematical Games column in Scientific American, he posed the following problem: "What is the smallest number of [queens] you can put on a board of side n such that no [queen] can be added without creating three in a row, a column, or a diagonal?" We use the Combinatorial Nullstellensatz to prove that this number is at least n, except in the case when n is congruent to 3 modulo 4, in which case one less may suffice. A second, more elementary proof is also offered in the case that n is even.Comment: 11 pages; lower bound in main theorem corrected to n-1 (from n) in the case of n congruent to 3 mod 4, minor edits, added journal referenc

Seasonal variation in reproductive condition of the Peco Bluntnose Shiner (\u3ci\u3eNotropis simus pecosensis\u3c/i\u3e)

Reproductive strategies vary among freshwater fishes. Information on reproductive characteristics is important for water managers’ efforts to protect and recover imperiled species. We describe aspects of the reproductive ecology of Pecos Bluntnose Shiner (Notropis simus pecosensis). We determined gonadosomatic index, examined ovarian condition, and counted mature ova to determine the seasonal reproductive timing and duration of Pecos Bluntnose Shiner in the Pecos River, New Mexico. Monthly changes in gonadosomatic index, ovarian stage, and number of mature ova per female suggest that Pecos Bluntnose Shiner has a reproductive season extending from April through September, with a peak occurring in June and July. In July, 93% (n = 44) of female Pecos Bluntnose Shiner were in reproductive condition, regardless of size. The highest number of mature ova counted (1498) was observed in a 59.5-mm (standard length) female from June 2009. Peak spawning activity coincides with sustained water releases from reservoirs; these releases have higher peaks and longer duration than natural rainfall events. Results of this study provide important insight on the reproductive biology of a threatened fish and may be useful in long-term conservation planning.Las estrategias reproductivas varían entre los peces de agua dulce. La información sobre las características reproductivas es importante para los gestores de los ambientes acuáticos con el fin de proteger y recuperar especies en peligro. Describimos aspectos de la ecología reproductiva de la carpa (Notropis simus pecosensis). Determinamos el índice gonadosomático, las condiciones del ovario, y contamos los óvulos maduros para determinar la temporada de reproducción y la duración de las carpas en el Río Pecos, Nuevo México. Los cambios mensuales en el índice gonadosomático, la etapa de ovario y el número de óvulos maduros por hembra sugirieron que las carpas tienen una temporada reproductiva desde abril hasta septiembre, con un pico entre junio y julio. En julio, el 93% (n = 44) de las hembras se encontraron en condiciones reproductivas, independientemente de su tamaño. El mayor número de óvulos maduros contabilizados (1498) en una hembra de 59.5 mm (longitud estándar) fue en junio del 2009. El pico máximo de actividad de desove coincide con una liberación prolongada de agua de los embalses, con picos mayores y mayor duración que los eventos de lluvia naturales. Los resultados de este estudio proporcionan información importante sobre la biología reproductiva de un pez amenazado y pueden ser útiles en la planificación de su conservación a largo plazo