Bonds intersecting long paths in $k$-connected graphs

TL;DR

This paper proves that in highly connected graphs, there exist minimal edge-cuts (bonds) intersecting all long paths, extending Gallai's question from vertices to bonds and providing bounds based on connectivity.

Contribution

It introduces the concept of bonds intersecting all long paths in k-connected graphs, generalizing previous vertex-based results and establishing new bounds depending on the graph's connectivity.

Findings

In 2-connected graphs, a bond intersects all paths of length at least p-1.

In 3-connected graphs, a bond intersects all paths of length at least p-2.

For k-connected graphs, bonds intersect all paths of length at least p-t+1, with t depending on k and p.

Abstract

A well-known question of Gallai (1966) asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex $v$ if and only if it meets the vertex-bond with respect to $v$. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let $p$ denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least $p-1$. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least $p-2$. For a $k$-connected graph $(k\ge3)$, we show that there is a bond meeting all paths of length at least $p-t+1$, where $t=\Big\lfloor\sqrt{\frac{k-2}{2}}\Big\rfloor$ if $p$ is even and $t=\Big\lceil\sqrt{\frac{k-2}{2}}\Big\rceil$ if $p$ is odd. Our results provide analogs of the corresponding results of P. Wu and S. McGuinness [Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.