Proper Rainbow Saturation in Graphs

Date

2024

Authors

Lane, Andrew

Journal Title

Journal ISSN

Volume Title

Publisher

University of Victoria

Abstract

This project explores graph theory, which is the study of networks (called “graphs”) with nodes called “vertices” and “edges” between pairs of vertices. A proper edge-colouring of a graph is an assignment of values to each edge of a graph such that edges that share a vertex receive different colours. This project specifically investigates the proper rainbow saturation problem, which is defined as follows: for a larger graph G and a smaller graph H, we say that G is properly rainbow H-saturated if G can be properly edge-coloured with no copy of H that has all different colours, but if any edge is added to G, then G always contains a copy of H with all different colours in any proper colouring. We seek to determine the proper rainbow saturation number, which is the minimum number of edges for G given a fixed graph H and a number of vertices for G. We improve on others’ results by finding exact values for the proper rainbow saturation number for specific graphs and proving general bounds for classes of graphs. This connects proper rainbow saturation to related graph saturation problems and reveals general patterns in the behaviour of these networks.

Description

Keywords

mathematics, graph theory, combinatorics, discrete mathematics, saturation, edge-colouring

Citation