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.     

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.     

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.     

Decomposing complete equipartite graphs into short even cycles

In this article we find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in the case that the length is both even and short relative to the number of parts. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:131‐143, 2011     

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.     

Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n ≥ 1, let G_n denote the complete graph of order n, and for all even integers n ≥ 2 let G_n denote the complete graph of order n with the edges of a 1‐factor removed. It is shown that for all non‐negative integers h and t and all positive integers n, G_n can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G_n. © 2004 Wiley Periodicals, Inc.     

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.     

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

We show that any complete -partite graph on vertices, with , whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours, given that the largest partition class of contains at most vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. From this we derive that the whole graph, if large enough, may be covered with 14 vertex-disjoint monochromatic cycles.     

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.