R&E 34

Catalogue of all
Book Series

List of Books
in this Series

Previous Book

Next Book

Research and Exposition in Mathematics -- Volume 34

   Enlarged Picture

Miroslav Haviar, Michal Ivaska

Vertex Labellings of Simple Graphs

ii+155 pages, soft cover, ISBN 978-3-88538-234-8, EUR 28.00, 2015

Graph theory belongs to the most dynamic disciplines within present mathematics, and due to its beauty and wide applications, also to the most popular areas of mathematics. The area of graph labellings, on whose selected part this monograph is focussing, is very young. Roughly speaking, a graph labelling is an assignment of integers to the vertices or edges, or both, of a graph, subject to certain conditions. The bases of the theory of graph labellings were laid out in the late 1960s, and since then a plethora of graph labellings methods and techniques have been studied in over 1900 research papers, monographs and theses.
One of the most famous open problems in graph theory nowadays is the Graceful Tree Conjecture which says that every tree can be gracefully labelled. A tree with m edges has a graceful labelling if its vertices can be assigned the labels 0,1,...,m such that the absolute values of the differences in vertex labels between adjacent vertices form the set {1,...,m}. The conjecture dates back to the 1960s and it is also known as the Ringel-Kotzig, Rosa or Ringel-Kotzig-Rosa Conjecture. Only limited progress has been made on the conjecture over the last fifty years despite numerous research papers and various theses and surveys.
This monograph adds new results and new approaches to the existing knowledge about vertex labellings of graphs. We believe that it brings advances in the study of vertex labellings of graphs and that it will be of interest to researchers in this area. We hope that the book will initiate further development in the study of the newly introduced concepts of Graph Chessboards and Labelling Relations as useful tools to investigate vertex labellings of graphs and to tackle the Graceful Tree Conjecture. We believe that especially the visualization provided via the graph chessboards, and the Graph Processor, developed and described in this book, could make this topic more accessible to working mathematicians as well as to students starting their research work in this area.

List of Contents