A tree version of Kőnig's theorem (Q1872889)

Let \(H\), \(F\) be families of subtrees of a tree \(T\). Let \(\sim\) be a symmetric relation on \(F\) containing the disjointness relation. If \(w(H,F,\sim)\) is the minimal size of a \(\sim\)-related subset of \(F\) which intersects every edge in \(H\), then \(w(H,F,\sim)\) equals the maximum of \(w(M,F,\sim)\) over all matchings \(M\) of \(H\). If \(H\) is a hypergraph on the union of disjoint sets \(X\) and \(Y\), \(T\) is a tree on \(Y\), every edge of \(H\) is the union of a singleton from \(X\) and a subtree of \(T\), then \(\sigma(H)\leq\nu(H)\) holds, where \(\sigma(H)\) is the minimal number \(n\) such that there are \(n\) edges \(e_1,\dots,e_n\) that \(W\) covers, where \(W\) is a union obtained the following way. We take either \(X\cap e_i\) or \(Y\cap e_i\) for each \(i\), independently of each other. \(\nu(H)\) is the size of a maximal matching in \(H\). If \(H\), \(F\) are hypergraphs, \(\text{iw}(H,F)\) is the minimal size of an \(H\)-covering matching in \(F\), \(\text{imw}(H,F)=\max \text{iw}(M,F)\) where the maximum is taken over all matchings \(M\) in \(H\), then it is proved that \(\text{imw}(H,H)=\text{iw}(H,H)\) when \(H\) is a hypergraph consisting of intervals of an ordered set. The matching witnessing equality is found by a polynomial algorithm. A special case concerning chordal graphs of Reed's conjecture on list colorings is derived.