Gray Codes from Antimatroids

Gara Pruesse, Department of Computer Science, University of Toronto.
Frank Ruskey, Department of Computer Science, University of Victoria.


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,F), let J(F) denote the graph whose vertices are the sets of F, where two vertices are adjacent if the corresponding sets differ by one element. Define J(F;k) to be the subgraph of J(F)2 induced by the sets in F with exactly k elements. Both graphs J(F) and J(F;k) are connected, and the former is bipartite.

We show that there is a Hamiltonian cycle in J(F) x K2. As a consequence, the ideals of any poset P may be listed in such a way that successive ideals differ by at most two elements. We also show that J(F;k) has a Hamilton path if (E,F) is the poset antimatroid of a series-parallel poset.

Similarly, we show that G(L) x K2 is Hamiltonian, where G(L) is the "basic word graph" of a language antimatroid (E,L). This result was known previously for poset antimatroids.

Keywords: Antimatroid, basic word graph, combinatorial Gray code, Hamiltonicity, join-distributive lattice.
AMS Classification: 05C45, 06A05, 06A06.

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