Fig. 1

Fully solved GCD network with χ=4, based on a small-world graph with N=20 nodes and 12 randomly added shortcuts (lower level), enriched by four leader nodes that form a complete subgraph (upper level). Between both levels, a total of 12 links connect the leader nodes with the ring. This figure illustrates the general principle of the graph coloring problem – finding a distribution of colors such that no same colors are linked (conflict-free coloring) and finding the minimal number of colors (the graph’s chromatic number) for which such a conflict-free coloring exists – as well as the distinction between ’ring nodes’ and ’leader nodes’ used throughout this investigation. For our investigation we assume the chromatic number to be given and study the convergence to a conflict-free coloring using dynamics generated by simple local update rules