Spanning trees of 3-uniform hypergraphs

Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r − 1) edges. For r = 3, Bollobás, Reed and Thomason proved the lower bound (1− 1/e)m/3 ≈ 0.21m, which was improved to 5m/9 by Bollobás and Scott (while the conjectured bound is 3m/5). In this paper, we show that this Bollobás-Thomason ...

On Steiner trees and minimum spanning trees in hypergraphs

Spanning trees in complete uniform hypergraphs and a connection to extended Shi hyperplane arrangements

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-Parking functions. We observe a simp...

Spanning trees in complete uniform hypergraphs and a connection to r-extended Shi hyperplane arrangements

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees of the complete graph on (n + 1) vertices and Parking functions of length n, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-P...

Matchings in 3-uniform hypergraphs

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose thatH is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than ( n−1 2 )