A Tight Bound for the Number of Edges of Matchstick Graphs

Abstract

AbstractA matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is $$\lfloor 3n-\sqrt{12n-3}\rfloor $$ ⌊ 3 n - 12 n - 3 ⌋ . In this paper we prove this conjecture for all $$n\ge 1$$ n ≥ 1 . The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.