Given a graph G=(V,E), a vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind et al. (1997)  studied proper t-frugal colourings and Yuster (1998)  studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. © 2010 Elsevier B.V. All rights reserved.
Kang, R. J., & Müller, T. (2011). Frugal, acyclic and star colourings of graphs. In Discrete Applied Mathematics (Vol. 159, pp. 1806–1814). https://doi.org/10.1016/j.dam.2010.05.008