A lower bound for constant-size local certification

Given a network property or data structure, local certification is labeling that allows to efficiently check the satisfied, structure correct. The quality of measured by size its labels: smaller, better. This notion plays central role in self-stabilization, because lower bound (and often an upper bound) on memory needed for silent self-stabilizing construction distributed structures. When it comes labels, one can identify three important regimes: properties which optimal polynomial number vertices graph, ones require only polylogarithmic size, and be certified with constant bits. first two regimes are well studied, several bounds, specific techniques, active research questions. On other hand, regime has never been really explored. main contribution this paper non-trivial regime. More precisely, we show using just bit (a binary certification), cannot certify k-colorability k≥3. To do so, develop new technique, based score, both symmetry arguments global parity argument. We hope technique will useful establishing stronger results. complement result related problem, illustrating some cases better than natural bound.