Bonds intersecting long paths in $k$-connected graphs

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.

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