Finding happiness: an analysis of the maximum happy vertices problem

The maximum happy vertices problem involves determining a vertex colouring of a graph such that the number of vertices assigned to the same colour as all of their neighbours is maximised. This problem is trivial if no vertices are precoloured, though in general it is NP-hard. In this paper we derive a number of upper and lower bounds on the number of happy vertices that are achievable in a graph and then demonstrate how certain problem instances can be broken up into smaller subproblems. Four different algorithms are also used to investigate the factors that make some problem instances more difficult to solve than others. In general, we find that the most difficult problems are those with relatively few edges and/or a small number of precoloured vertices. Ideas for future research are also discussed.