arXiv subjects
Publications and source records attributed to Arthur Ulmer.
Menger's theorem on $A$--$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.
Let $\Gamma$ be an Abelian group. In this paper I characterize the $A$-paths of weight $0\in\Gamma$ that have the Erd\H{o}s-P\'osa property. Using this in an auxiliary graph, one can also easily characterize the $A$-paths of weight $\gamma\in\Gamma$ that have the Erd\H{o}s-P\'osa property. These results also extend to long paths, that is paths of some minimum length. A structural result on zero walls with non-zero linkages will be proven as a means to prove the main result of this paper. This immediately implies that zero cycles with respect to an Abelian group $\Gamma$ have the Erd\H{o}s-P\'osa property.
We prove that the ladder with $3$~rungs and the house graph have the edge-Erd\H{o}s-P\'{o}sa property, while ladders with $14$~rungs or more have not. Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit a better result.
We show that A-paths of length 0 modulo 4 have the Erdős-Pósa property. We also prove that A-paths of length 2 modulo 4 have the property but that A-paths of length 1 or of length 3 modulo 4 do not have it.
A classic theorem of Erd\H{o}s and P\'osa (1965) states that every graph has either $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices meeting all its cycles. While the standard proof revolves around finding a large `frame' in the graph (a subdivision of a large cubic graph), an alternative way of proving this theorem is to use a ball packing argument of K\"uhn and Osthus (2003) and Diestel and Rempel (2005). In this paper, we argue that the latter approach is particularly well suited for studying edge variants of the Erd\H{o}s-P\'osa theorem. As an illustration, we give a short proof of a theorem of Bruhn, Heinlein, and Joos (2019), that cycles of length at least $\ell$ have the so-called edge-Erd\H{o}s-P\'osa property. More precisely, we show that every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\ell$ or an edge set $F$ of size $O(k\ell \cdot \log (k\ell))$ such that $G-F$ has no cycle of length at least $\ell$. For fixed $\ell$, this improves on the previously best known bound of $O(k^2 \log k +k\ell)$.
For a fixed integer $\ell$ a path is long if its length is at least $\ell$. We prove that for all integers $k$ and $\ell$ there is a number $f(k,\ell)$ such that for every graph $G$ and vertex sets $A,B$ the graph $G$ either contains $k$ edge-disjoint long $A$-$B$-paths or it contains an edge set $F$ of size $|F|\leq f(k,\ell)$ that meets every long $A$-$B$-path. This is the edge analogue of a theorem of Montejano and Neumann-Lara (1984). We also prove a similar result for long $A$-paths and long $\mathcal{S}$-paths.
In this short note we prove that for every $k\in \mathbb{N}$ there is a $t_k\in\mathbb{N}$ such that for every digraph $G$ there are either $k$ edge-disjoint directed cycles in $G$ or a set $X$ of at most $t_k$ edges such that $G-X$ contains no directed cycle.