Long spirals

Investigation of spiral curves by exploring the properties related to the way of their winding is continued. In particular, W. Vogt’s theorem concerning boundary angles of a spiral arc is further extended. Bibliography: 3 titles.     

Short spirals

Previous results for planar curves having monotone curvature (spirals) and limited in length in one way or another are extended by dropping the requirements of convexity of the arc, injectivity of the projection onto the spanning chord, and continuity of the curvature. A more general class of short spiral arcs is thus introduced. The results include W. Vogt’s theorem concerning boundary angles of a convex spiral arc, necessary and sufficient conditions for its existence, and positional inequalities for such arcs. Bibliography: 8 titles.     

Homomorphisms of graphs into odd cycles

We give a class of graphs G for which there exists a homomorphism (= adjacency preserving map) from V(G) to V(C), where C is the shortest odd cycle in G, thereby extending a result of Albertson, Catlin, and Gibbons. Our class of graphs is characterized by the following property: For each odd subdivision G′ of G there exists a homomorphic map from V(G′) to V(C), where C′ is the shortest odd cycle of G′.     

Turning Euler's Factoring Method into a Factoring Algorithm

An algorithm is presented which, given a positive integer n,will either factor n or prove it to be prime. The algorithmtakes O(n^1/3+) steps.     

Decompositions of digraphs into paths and cycles

It is shown that in a digraph G, there is an elementary directed path or an elementary directed cycle meeting all inclusion-maximal demi-cocycles of G. This theorem is used to obtain an upper bound on the cardinality of a minimal partition of the arc set of G into directed paths and cycles.     

偶长圈加一条弦的分拆

设Kv是一个v点完全图.G是一个有限简单图.Kv上的一个图设计G-GD是一个对子（X,B）,其中X是Kv的顶点集合,B是Kv的一些与G同构的子图（称为区组）的集合,使得Kv的任意一条边恰出现在B的一个区组中.文中讨论的简单图是C^（r）2k,即带有一条弦的2k长圈,其中r表示弦的两个端点之间的顶点个数,1≤r≤k-1.文中给出了一个构作C^（r）m设计的统一方法,并得到关于v≡0,1（mod2k＋1）时C^（r）2k-GD（v）的一系列结果.     

Partitioning edge-coloured complete graphs into monochromatic cycles

A conjecture of Erdös, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for $r=2$. In this note we show that in fact this conjecture is false for all $r⩾3$. We also discuss some weakenings of this conjecture which may still be true.     

A certain feature of logarithmic spirals

A curve formed by inversion of a logarithmic spiral is called a double logarithmic spiral. The curves in this family possess the following property: there always exists such a spiral with continuous and monotone curvature satisfying any possible boundary conditions (endpoints, tangents, and curvatures). The problem of constructing a spiral with continuous curvature and prescribed curvature elements at the endpoints is thus solved. Bibliography: 6 titles.     

Decomposing the complete graph into cycles of many lengths

For all oddv 3 the complete graph onvK _v vertices can be decomposed intov – 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v – 1. Also, for all oddv 7,K _v can be decomposed intov – 3 edge disjoint cycles whose lengths are 3, 4,...,v – 4,v – 2,v – 1,v. Research supported by Australian Research Council grant A49130102     

Factorizations of product graphs into cycles of uniform length

The following question is raised by Alspach, Bermond and Sotteau: IfG ₁ has a decomposition into hamilton cycles and a 1-factor, andG ₂ has a hamilton cycle decmposition (HCD), does their wreath productG ₁ *G ₂ admit a hamilton cycle decomposition? In this paper the above question is answered with an additional condition onG ₁. Further it is shown that some product graphs can be decomposed into cycles of uniform length, that is, the edge sets of the graphs can be partitioned into cycles of lengthk, for some suitablek.     

Construction of spirals with prescribed boundary conditions

Spirality, regarded as monotonicity of curvature, is preserved under inversions. This property is used for constructing a spiral transition curve with predefined curvature elements at the endpoints. These boundary conditions define two invariant values: Coxeter’s inversive distance and the width of the lens. In order to solve the problem, it suffices to realize the corresponding values on two curvature elements of any known spiral. The rest is achieved by inversion. In particular, any boundary conditions compatible with spirality can be satisfied by inverting an arc of the logarithmic spiral. Bibliography: 9 titles.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.     

Archimedes and the spirals: The heuristic background

In his work, The Method, Archimedes displays the heuristic technique by which he discovered many of his geometric theorems, but he offers there no examples of results from Spiral Lines. The present study argues that a number of theorems on spirals in Pappus' Collectio are based on early Archimedean treatments. It thus emerges that Archimedes' discoveries on the areas bound by spirals and on the properties of the tangents drawn to the spirals were based on ingenious constructions involving solid figures and curves. A comparison of Pappus' treatments with the Archimedean proofs reveals how a formal stricture against the use of solids in problems relating exclusively to plane figures induced radical modifications in the character of the early treatments.     

Planar interpolation with a pair of rational spirals

Spirals are curves of one-signed, monotone increasing or decreasing curvature. Spiral segments have the advantage that the minimum and maximum curvatures are at their endpoints. Two situations where the planar, two-point G²$G^2$ Hermite interpolation problem can be solved with a pair of rational spiral segments are outlined here.     

Stability of a functional equation for square root spirals

By using an idea of Heuvers, Moak and Boursaw [1], we will prove a Hyers-Ulam-Rassias stability (or a general Hyers-Ulam stability) of the functional equation (1), which is closely related to the square root spiral.     

3D Euler spirals for 3D curve completion

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curve, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of this curve - properties that have been shown to be important for the appeal of curves. We illustrate its utility in two applications. The first is “fixing” curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.