Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence $v_1,e_1,v_2,\dots,v_n,e_n$ of distinct vertices $v_i$ and distinct edges $e_i$ so that each $e_i$ contains $v_i$ and $v_{i+1}$. So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever $k \ge 4$ and $n \ge 30$. Our argument is based on the Kruskal-Katona theorem. The case when $k=3$ was already solved by Verrall, building on results of Bermond

Berge Sorting

In 1966, Claude Berge proposed the following sorting problem. Given a string of $n$ alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, rearrange the pegs into a string consisting of $\lceil\frac{n}{2}\rceil$ white pegs followed immediately by $\lfloor\frac{n}{2}\rfloor$ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ such {\em Berge 2-moves} for $n\geq 5$. Extending Berge's original problem, we consider the same sorting problem using {\em Berge $k$-moves}, i.e., moves which take $k$ adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge 3-moves for $n\not\equiv 0\pmod{4}$ and in $\lceil\frac{n}{2}\rceil+1$ Berge 3-moves for $n\equiv 0\pmod{4}$, for $n\geq 5$. In general, we conjecture that, for any $k$ and large enough $n$, the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge $k$-moves. This estimate is tight as $\lceil\frac{n}{2}\rceil$ is a lower bound for the minimum number of required Berge $k$-moves for $k\geq 2$ and $n\geq 5$.Comment: 10 pages, 2 figure

Non-simple genus minimizers in lens spaces

Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non-simple genus minimizers. This negatively answers a question of Rasmussen.Comment: 16 pages, 6 figure

Linearity of Saturation for Berge Hypergraphs

For a graph F, we say a hypergraph H is Berge-F if it can be obtained from F be replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of Berge-F. The k-uniform saturation number of Berge-F, satk(n, Berge-F) is the fewest number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that satk(n, Berge-F) = O(n) for all graphs F and uniformities 3 ≤ k ≤ 5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraph

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries.Comment: 27 pages, 8 figures; Expositional improvements, corrected normalization of A grading in proof of Lemma 4.1

Generalizations of Grillet's theorem on maximal stable sets and maximal cliques in graphs

AbstractGrillet established conditions on a partially ordered set under which each maximal antichain meets each maximal chain. Berge pointed out that Grillet's theorem can be stated in terms of graphs, made a conjecture that strengthens it, and asked a related question. We exhibit a counterexample to the conjecture and answer the question; then we prove four theorems that generalize Grillet's theorem in the spirit of Berge's proposals