A minimum-change version of the Chung-Feller theorem for Dyck paths

A Dyck path with 2k steps and e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1, 1) or (1,−1), respectively, and that has exactly e many ↘-steps below the line y = 0. Denoting by D 2k the set of Dyck paths with 2k steps and e flaws, the Chung-Feller theorem asserts that the sets D 2k, D 1 2k, . . . , D k 2k all have the same cardinality 1 k+1 ( 2k k ) = Ck, the k-th Catalan number. The standard combinatorial proof of this classical result establishes a bijection f ′ between D 2k and D e+1 2k that swaps certain parts of the given Dyck path x, with the effect that x and f ′(x) may differ in many positions. In this paper we strengthen the Chung-Feller theorem by presenting a simple bijection f between D 2k and D e+1 2k which has the additional feature that x and f(x) differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of f in constant time per generated Dyck path. As an application, we use our minimum-change bijection f to construct cycle-factors in the odd graph O2k+1 and the middle levels graph M2k+1 — two intensively studied families of vertex-transitive graphs — that consist of Ck many cycles of the same length.