An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Developing a Mathematical Model for Bobbin Lace

Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as Computer Scientists and Mathematician

Structure and enumeration of (3+1)-free posets

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to clarify the construction of skeleta and the enumeration. An extended abstract of this paper appears as arXiv:1212.535

Enumerating Hamiltonian Cycles in A 2-connected Regular Graph Using Planar Cycle Bases

Planar fundamental cycle basis belong to a 2-connected simple graph is used for enumerating Hamiltonian cycles contained in the graph. This is because a fun- damental cycle basis is easily constructed. Planar basis is chosen since it has a weighted induced graph whose values are limited to 1. Hence making it is possible to be used in the Hamiltonian cycle enumeration procedures efficiently. In this paper a Hamiltonian cycle enumeration scheme is obtained through two stages. Firstly, i cycles out of m bases cycles are determined using an appropriate con- structed constraint. Secondly, to search all Hamiltonian cycles which are formed by the combination of i basis cycles obtained in the first stage efficiently. This ef- ficiency is achieved through the generation of a class of objects consisting of Ill-bit binary strings which is a representation of i cycle combinations between m cycle basis cycle

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original