Color the cycles

The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013

The Ramsey number of loose paths in 3-uniform hypergraphs

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$

Electron Population Aging Models for Wide-Angle Tails

Color-color diagrams have been useful in studying the spectral shapes in radio galaxies. At the workshop we presented color-color diagrams for two wide-angle tails, 1231+674 and 1433+553, and found that the standard aging models do not adequately represent the observed data. Although the JP and KP models can explain some of the observed points in the color-color diagram, they do not account for those found near the power-law line. This difficulty may be attributable to several causes. Spectral tomography has been previously used to discern two separate electron populations in these sources. The combination spectra from two such overlying components can easily resemble a range of power-laws. In addition, any non-uniformity in the magnetic field strength can also create a power-law-like spectrum. We will also discuss the effects that angular resolution has on the shape of the spectrum.Comment: 4 pages, 1 figure, proceedings from 1999 'Life Cycles of Radio Galaxies' workshop at STScI in Baltimore, M

Color and Translucency Variation of a One-Shaded Resin-Based Composite after Repeated Heating Cycles and Staining

Background: This study aimed to determine the effect of repeated pre-polymerization heating on the color and translucency of a one-shaded resin-based composite and to evaluate whether the heating cycles affect its color stability. (2) Methods: Fifty-six samples of 1-mm thickness were fabricated from Omnichroma (OM) after applying different heating cycles (for one, five, and ten times at 45 C) before polymerization (n = 14/group) and afterwards were stained with a yellow dye solution. CIE L*, a*, b*, C*, h coordinates were recorded, and color differences, whiteness, and translucency were calculated, before and after staining. (3) Results: Heating cycles significantly influenced the color coordinates, WID00, and TP00 of OM being higher after one heating cycle and decreasing as the number of heating cycles increased. The color coordinates, WID, and TP00 after staining significantly differed for each group. The color and whiteness differences calculated after staining exceeded the acceptability thresholds for all groups. The color and whiteness variations after staining were clinically unacceptable. (4) Conclusions: Repeated pre-polymerization heating induces a clinically acceptable color and translucency change to OM. Although the color changes resulting after staining are clinically unacceptable, increasing the number of heating cycles up to ten times slightly reduces the color differences.Research Project PCD 1032/55 from the Iuliu Hatieganu University of Medicine and Pharmacy, Cluj-Napoca, Romani

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time $\tau_{2\sigma}$ for the random graph process to have minimum degree $2\sigma$ coincides with the hitting time for $\sigma$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $\sigma\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $\sigma$ colors so that at time $\tau_{2\sigma}$, each color class is Hamiltonian.Comment: Minor change