Graphs and graph polynomials

• Main Author: Kriel, Christo
• Format: Others
• Language: en
• Published: 2018
• Online Access: https://hdl.handle.net/10539/25012

A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017 === In this work we study the k-defect polynomials of a graph G. The k defect polynomial is a function in λ that gives the number of improper colourings of a graph using λ colours. The k-defect polynomials generate the bad colouring polynomial which is equivalent to the Tutte polynomial, hence their importance in a more general graph theoretic setting. By setting up a one-to-one correspondence between triangular numbers and complete graphs, we use number theoretical methods to study certain characteristics of the k-defect polynomials of complete graphs. Specifically we are able to generate an expression for any k-defect polynomial of a complete graph, determine integer intervals for k on which the k-defect polynomials for complete graphs are equal to zero and also determine a formula to calculate the minimum number of k-defect polynomials that are equal to zero for any complete graph. === XL2018