We investigate the problem of packing and covering odd (u,v)-trails in a
graph. A (u,v)-trail is a (u,v)-walk that is allowed to have repeated
vertices but no repeated edges. We call a trail odd if the number of edges in
the trail is odd. Let ν(u,v) denote the maximum number of edge-disjoint odd
(u,v)-trails, and Ï„(u,v) denote the minimum size of an edge-set that
intersects every odd (u,v)-trail.
We prove that τ(u,v)≤2ν(u,v)+1. Our result is tight---there are
examples showing that τ(u,v)=2ν(u,v)+1---and substantially improves upon
the bound of 8 obtained in [Churchley et al 2016] for τ(u,v)/ν(u,v).
Our proof also yields a polynomial-time algorithm for finding a cover and a
collection of trails satisfying the above bounds.
Our proof is simple and has two main ingredients. We show that (loosely
speaking) the problem can be reduced to the problem of packing and covering odd
(uv,uv)-trails losing a factor of 2 (either in the number of trails found, or
the size of the cover). Complementing this, we show that the
odd-(uv,uv)-trail packing and covering problems can be tackled by exploiting
a powerful min-max result of [Chudnovsky et al 2006] for packing
vertex-disjoint nonzero A-paths in group-labeled graphs