Oddness to resistance ratios in cubic graphs

Let $G$ be a bridgeless cubic graph. Oddness (weak oddness) is defined as the minimum number of odd components in a 2-factor (an even factor) of $G$, denoted as $\omega \left(G\right)$ (Steffen, 2004) (${\omega }^{\prime }\left(G\right)$ Lukot’ka and Mazák (2016)). Oddness and weak oddness have been referred to as measurements of uncolourability (Fiol et al., 2017, Lukot’ka and Mazák, 2016, Lukot’ka et al., 2015 and, Steffen, 2004), due to the fact that $\omega \left(G\right)=0$ and ${\omega }^{\prime }\left(G\right)=0$ if and only if $G$ is 3-edge-colourable. Another so-called measurement of uncolourability is resistance, defined as the minimum number of edges that can be removed from $G$ such that the resulting graph is 3-edge-colourable, denoted as $r\left(G\right)$ (Steffen, 2004). It is easily shown that $\omega \left(G\right)\ge {\omega }^{\prime }\left(G\right)\ge r\left(G\right)$. While it has been shown that the difference between any two of these measures can be arbitrarily large, it has been conjectured that ${\omega }^{\prime }\left(G\right)\le 2r\left(G\right)$, and that if $G$ is a snark then $\omega \left(G\right)\le 2r\left(G\right)$ (Fiol et al., 2017). In this paper, we disprove the latter by showing that the ratio of oddness to weak oddness can be arbitrarily large. We also offer some insights into the former conjecture by defining what we call resistance reducibility, and hypothesizing that almost all cubic graphs are such resistance reducible.     

Monotonicity properties of ‐functions

We present some monotonicity results for a class of Dirichlet series generalizing previously known results. The fact that is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Lastly, we use similar techniques to prove another formulation of the Riemann hypothesis for the L‐function associated to the Ramanujan‐tau function.     

Extending partial representations of circle graphs

The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph and a partial representation giving some predrawn chords that represent an induced subgraph of . The question is whether one can extend to a representation of the entire graph , that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of . Our main result is an time algorithm for partial representation extension of circle graphs, where is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.     

Specific features of complete-absorption measurement in selective intracavity laser spectroscopy

For multimode pulsed dye lasers (of nanosecond and microsecond duration with frequency-dependent losses in the cavity), experimental studies and theoretical calculations have been made of the dependence of the value of the equivalent line width, used as a measure of selective absorption, on the main parameters characterizing the laser, the cavity, and the absorbent. Analytical relations have been obtained which describe the main aspects of complete absorption in lines with dispersion, Gaussian, or mixed spectra profiles, and their experimental testing has been carried out. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 3, pp. 327–332, May–June, 2000.     

Word绘图功能在物理实验教学活动中的应用

本介绍Wierosoft公司Office组件Word的绘图功能在物理实验方面的应用。在掌握Word绘图基本的方法及技巧后。能胜任物理实验教学实践活动中的作图要求。     

Discrepancy of random graphs and hypergraphs

Answering in a strong form a question posed by Bollobás and Scott, in this paper we determine the discrepancy between two random k‐uniform hypergraphs, up to a constant factor depending solely on k. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 47, 147–162, 2015