Fast Distributed Coloring Algorithms for Triangle-Free Graphs

Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree- ∆ graphs may require palettes of ∆+1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find (∆/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees, where k is at most ( 1 4 − o(1)) ln ∆ in triangle-free graphs and at most (1 − o(1)) ln ∆ in girth-5 graphs and trees, and o(1) is a function of ∆. Specifically,for∆sufficiently large we can find such a coloring in O(k +log ∗ n)time.Moreover,forany ∆ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O(log ∆ +log∆log n) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most (4 + o(1)) ∆ ln ∆, which improves on Jamall’s recent bound of (67 + o(1)) ∆ ln ∆ 1. Also, we show that (∆+1)-coloring for triangle-free graphs can be obtained in sublogarithmic time for any ∆

Atributos físicos de um planossolo háplico sob sistemas de manejo comparados aos do campo nativo Physical attributes of an albaqualf under management systems in comparison with those under native grassland

Sistemas de manejo podem influenciar as condições físico-hídricas do solo e a produtividade das culturas. O objetivo deste estudo foi avaliar os atributos físicos de um Planossolo Háplico sob diferentes sistemas de manejo e compará-los com os encontrados no campo nativo. A densidade, a condutividade hidráulica de solo saturado, a porosidade e a agregação foram avaliadas nas camadas de 0,000-0,025; 0,025-0,075; 0,075-0,125; 0,125-0,175; 0,175-0,275 m. O sistema plantio direto e o preparo convencional apresentaram condições físico-hídricas similares, enquanto o solo sob campo nativo proporcionou condições adequadas ao crescimento e desenvolvimento de plantas por apresentar maior condutividade hidráulica, macroporosidade, porosidade total e agregação, e menor densidade do solo. Sugerem-se valores críticos para as plantas, respectivamente de densidade, de condutividade hidráulica saturada, de diâmetro médio ponderado de agregados de 1,56 Mg m-3; 16,18 mm h-1 e 2,49 mm.<br>The soil management may influence soil hydro-physical conditions and crop yield. The objective of this study was to determine soil physical properties of an Albaqualf under management systems and to compare them with those under a native pasture. The bulk density, saturated hydraulic conductivity, macroporosity, microporosity, total porosity and aggregation were evaluated in the soil layers 0-0.025; 0.025-0.075; 0.075-0.125; 0.125-0.175; and 0.175-0.275 m. The hydro-physical soil conditions in no-till and conventional systems were similar. The conditions in the soil under native grassland were more adequate for crop development due to a higher hydraulic conductivity, macroporosity, total porosity and aggregation and lower bulk density. Critical values considered adequate for the crop development of 1.56 Mg m-3; 16.18 mm h-1 and 2.49 mm, respectively, for bulk density, saturated hydraulic conductivity and mean weight diameter were suggested