Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...

We consider an analogue of the well-known Casson knot invariant for knotoids. start with a direct classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value extension is formal sum subgroups first group $$H_1(\varSigma )$$ where $$\varSigma $$ oriented surface (maybe) non-empty boundary in diagrams lie. To make informat...

In a storyline visualization, we visualize a collection of interacting characters (e.g., in a movie, play, etc.) by x-monotone curves that converge for each interaction, and diverge otherwise. Given a storyline with n characters, we show tight lower and upper bounds on the number of crossings required in any storyline visualization for a restricted case. In particular, we show that if (1) each ...

[1] The present empirical models describing a location and shape of the magnetopause and bow shock are based on a statistical evaluation of magnetopause and bow shock crossings. The crossings are usually identified by a visual inspection of the plots or by automatic methods which are less reliable. We present a new method of determination of the most probable boundary locations. The method is b...

Many methods have been used to survey red fox (Vulpes vulpes) and gray fox (Urocyon dnereoargenteus) populations. However, none has proven entirely satisfactory, and wild foxes remain one of the most difficult economically important wildlife species to monitor. In this study we evaluated the reliability of a snow track survey method for foxes by investigating whether the average number of road ...

Railroad-highway grade crossings present unique safety and operational challenges on both the rail and highway side. Consolidation of these crossings can be a valuable tool to improve safety, but is often a source of disagreement among interested parties. In order to better evaluate crossings to determine their suitability to be consolidated, a spreadsheet-based tool was developed to rank all p...

The calculation of the Floquet quasi-energies of a system driven by a time-periodic field is an efficient way to understand its dynamics. In particular, the phenomenon of dynamical localization can be related to the presence of close approaches between quasienergies ~either crossings or avoided crossings!. Here we consider a driven two-level system and study how the locations of crossings in th...

Contour crossings are a form of numerical error common to contour dynamical and contour-advective simulations of geophysical fluid flows. This thesis is devoted to the study of contour crossings that occur in simulations of fluid motion performed by the Contour-Advective Semi-Lagrangian (CASL) algorithm. Components of the CASL algorithm relevant to the study of contour crossings are discussed. ...

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that ≤ m -reduce to NP sets. A natural variant accepts exactly the languages that ≤ m -reduce to P sets. We show that these two classes coincide if and only if P = P, where the latter class denotes the sets acceptable via at most one question to #P followed by at...

In this paper we study the enumeration of diagrams of n chords joining 2n points on a circle in disjoint pairs. We establish limit laws for the following three parameters: number of components, size of the largest component, and number of crossings. We also nd exact formulas for the moments of the distribution of number of components and number of crossings. Abstract In this paper we study the ...