Jump-systems of $T$-paths

Abstract

Jump systems are sets of integer vectors satisfying a simple axiom, generalizing matroids, also delta-matroids, and well-kown combinatorial examples such as degree sequences of subgraphs of a graph. It is useful to know if a set of vectors defined from combinatorial structures is a jump system: this has consequences for optimizing on the set, or on some derived sets of vectors. In this note we are mainly concerned in telling our proof of the following more than two decades old fact and its original, elementary proof for an example different from degree sequences: {\em Given an udirected graph $G=(V,E)$ and $T\subseteq V$, the vectors $m$ indexed by $T$ for which there exist a set of openly disjoint $T$-paths so that each $t\in T$ is the endpoint of exactly $m(t)$ paths forms a jump system. The same holds for edge-disjoint $T$-paths.} We are also exhibiting the context and some consequences of this fact, with some pointers to recent developments, among them ro another proof by Iwata and Yokoi, to some related jump system intersection theorems and to some open problems.