From Coin Play to Euler's Formula: A Contact-Graph Characterization and Existence Question for Hole-Tight Disk Arrangements

This revised preprint investigates the relationship between contact graphs and the number of bounded holes in non-overlapping disk arrangements, motivated by configurations of coins. For a connected arrangement of n disks, the number H of bounded complementary components satisfies H = E - n + 1, where E is the number of pairwise tangencies. Defining a hole-tight arrangement through the independent contact condition E = 2n - 3 yields the equivalence that an arrangement is hole-tight if and only if H = n - 2. The paper establishes the existence of hole-tight arrangements for arbitrary positive radii when n = 2 or 3, and for arbitrary n ≥ 2 when all disk radii are equal. The general existence question for arbitrary prescribed, possibly unequal radii is explicitly presented as a conjecture rather than a proven theorem. An explicit four-disk construction distinguishes the existence of an arrangement with n - 2 holes from the stronger and generally false assertion that n - 2 is the global maximum number of holes. Preprint status: This manuscript is a revised research preprint and has not been certified by peer review. It substantively corrects the mathematical formulation of an earlier version submitted to Discrete & Computational Geometry. The revised preprint does not replace the manuscript currently held by the journal.