The Hypergraphic Tutte/Nash-Williams Theorem via Integer Decomposition, Total Dual Laminarity, and Power Matroids

We reprove the hypergraphic generalization of the Tutte/Nash-Williams theorem, which gives sufficient conditions for a hypergraph to contain k disjoint connected hypergraphs. First we observe the theorem is equivalent to the natural LP relaxation having the integer decomposition property. Then we give a new proof of this property using LP uncrossing methods. We discover that “total dual laminarity” precisely characterizes g-polymatroids, and we note that hypergraphic matroids can be viewed as a special case of “power matroids.”