Harmonious colourings of temporal matchings

Abstract

Graph colouring is a fundamental problem in computer science, with a large body of research dedicated to both the general colouring problem and restricted cases. Harmonious colourings are one such restriction, where each edge must contain a globally unique pair of colours, i.e. if an edge connects a vertex coloured x with a vertex coloured y, then no other pair of connected vertices can be coloured x and y. Finding such a colouring in the traditional graph setting is known to be NP-hard, even in trees. This paper considers the generalisation of harmonious colourings to Temporal Graphs, specifically $(k,t)$-Temporal matchings, a class of temporal graphs where the underlying graph is a matching (a collection of disconnected components containing pairs of vertices), each edge can appear in at most t timesteps, and each timestep can contain at most k other edges. We provide a complete overview of the complexity landscape of finding temporal harmonious colourings for $(k,t)$-matchings. We show that finding a Temporal Harmonious Colouring, a colouring that is harmonious in each timestep, is NP-hard for (k,t)-Temporal Matchings when $k\ge 2,t\ge 4$, or when $k\ge 3$ and $t\ge 2$. We further show that this problem is inapproximable for $t\ge 2$ and an unbounded value of k, and that the problem of determining the temporal harmonious chromatic number of a $(2,3)$-temporal matching can be determined in linear time. Finally, we strengthen this result by a set of upper and lower bounds of the temporal harmonious chromatic number both for individual temporal matchings and for the classes of $(k,t)$-temporal matchings, paths, and cycles.