l-Harmonious Graph Colouring
Lauren DeDieu · 2012
In 1983, Hopcroft and Krishnamoorthy defined a new type of graph colouring called harmonious colouring. Harmonious colouring is a proper vertex colouring such that no two edges share the same colour pair. The least number of colours needed to harmoniously colour a graph is called the harmonious chromatic number. We will examine the results found for the harmonious chromatic number of paths, cycles, and trees. We will also extend the definition of harmonious colouring and define λ-harmonious colouring, which allows each edge colour pair to occur up to λ times. We will explore λ-harmonious colouring and will prove some results for the λ-harmonious chromatic number of complete graphs, complete bipartite graphs, paths, cycles, and wheels.