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A Dyck path with $2k$ steps and $e$ flaws is a path in the integer latticethat starts at the origin and consists of $k$ many $\nearrow$-steps and $k$many $\searrow$-steps that change the current coordinate by $(1,1)$ or$(1,-1)$, respectively, and that has exactly $e$ many $\searrow$-steps belowthe line $y=0$. Denoting by $D_{2k}^e$ the set of Dyck paths with $2k$ stepsand $e$ flaws, the Chung-Feller theorem asserts that the sets$D_{2k}^0,D_{2k}^1,\ldots,D_{2k}^k$ all have the same cardinality$\frac{1}{k+1}\binom{2k}{k}=C_k$, the $k$-th Catalan number. The standardcombinatorial proof of this classical result establishes a bijection $f'$between $D_{2k}^e$ and $D_{2k}^{e+1}$ that swaps certain parts of the givenDyck path $x$, with the effect that $x$ and $f'(x)$ may differ in manypositions. In this paper we strengthen the Chung-Feller theorem by presenting asimple bijection $f$ between $D_{2k}^e$ and $D_{2k}^{e+1}$ which has theadditional feature that $x$ and $f(x)$ differ in only two positions (the leastpossible number). We also present an algorithm that allows to compute asequence of applications of $f$ in constant time per generated Dyck path. As anapplication, we use our minimum-change bijection $f$ to construct cycle-factorsin the odd graph $O_{2k+1}$ and the middle levels graph $M_{2k+1}$ --- twointensively studied families of vertex-transitive graphs --- that consist of$C_k$ many cycles of the same length.

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