Packing and Covering Odd (u,v)-trails in a Graph

Abstract

In this thesis, 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 \emph{odd} if the number of edges in the trail is odd. Given a graph $G$ and two specified vertices $u$ and $v$, the odd $(u,v)$-trail packing number, denoted by $\nu(u,v)$, is the maximum number of edge-disjoint odd $(u,v)$-trails in $G$. And, the odd $(u,v)$-trail covering number, denoted by $\tau(u,v)$, is the minimum size of an edge-set that intersects every odd $(u,v)$-trail in $G$. In 2016, Churchley, Mohar, and Wu, were the first ones to prove a constant factor bound on the \coverpack ratio, by showing that $\tau(u,v) \leq 8 \cdot \nu(u,v)$. Our main result in this thesis is an improved bound on the covering number: $\tau(u,v) \leq 5 \cdot \nu(u,v) + 2$. The proof leads to a polynomial-time algorithm to find, for any given $k \geq 1$, either $k$ edge-disjoint odd $(u,v)$-trails in $G$ or a set of at most $5k-3$ edges intersecting all odd $(u,v)$-trails in $G$.