Gray codes from antimatroids

Abstract

We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids.

For antimatroid (E,$\mathcal{F}$), letJ($\mathcal{F}$) denote the graph whose vertices are the sets of$\mathcal{F}$, where two vertices are adjacent if the corresponding sets differ by one element. DefineJ($\mathcal{F}$;k) to be the subgraph ofJ($\mathcal{F}$)² induced by the sets in$\mathcal{F}$ with exactlyk elements. Both graphsJ($\mathcal{F}$) andJ($\mathcal{F}$;k) are connected, and the former is bipartite.

We show that there is a Hamiltonian cycle inJ($\mathcal{F}$)×K ₂. As a consequence, the ideals of any poset % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepuaaa!414C!\[\mathcal{P}\] may be listed in such a way that successive ideals differ by at most two elements. We also show thatJ($\mathcal{F}$;k) has a Hamilton path if (E,$\mathcal{F}$) is the poset antimatroid of a series-parallel poset.

Similarly, we show thatG($\mathcal{L}$)×K ₂ is Hamiltonian, whereG($\mathcal{L}$) is the “basic word graph” of a language antimatroid (E,$\mathcal{L}$). This result was known previously for poset antimatroids.