A linear-time algorithm for clique-coloring planar graphs
Abstract A clique of a graph G is a set of pairwise adjacent vertices of G . A clique-coloring of G is an assignment of colors to the vertices of G such that no inclusion-wise maximal clique of size at least 2 is monochromatic. Mohar and Skrekovski proved that planar graphs are 3-clique-colorable (Electr. J. Combin. 6 (1999), #R26). In this paper we present a linear time algorithm for 3-clique-coloring planar graphs by giving a new proof that planar graphs are 3-clique-colorable.
The specification of non-functional requirements, e. g., on timing forms an essential part of modern system design. Modeling languages such as MARTE/CCSL provide dedicated description means enabling engineers to formally define the ticking of the clocks to be implemented in terms of clock constraints and the actually intended timing behavior in terms of instant relations. But thus far, instant relations have only been utilized in order to monitor the correct execution of the clock constraints. In this work, we propose a methodology which, for the first time, verifies clock constraints against the given instant relations. To this end, the timing behavior is represented in terms of an automaton followed by its verification through satisfiability solvers. A case study illustrates the application of the proposed methodology.