In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this paper, we show that in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it gives a new proof for the result on split graphs, as split graphs are $2K_2$-free.
In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain $K_4$ as a minor. Series-parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present a proof that every connected series-parallel graph has a vertex that is co…
In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to Gallai's question for many well-known classes of graphs such as split graphs, series parallel graphs, and $2K_2$-free graphs. Among all the graph classes that support Gallai's question, almost all of them were shown to be Hamiltonian under certain conditions. This observation motivates us to investigate Gallai's question in graphs that are "close" to Hamiltonici…
We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in which the complement of the intersection of vertex sets $V(C)\cap V(D)$ of two longest paths is connected, is minimal.
We show that every pair of longest paths in a $k$-connected graph on $n$ vertices intersect each other in at least $(8k-n+2)/5$ vertices. We also show that, in a 4-connected graph, every pair of longest paths intersect each other in at least four vertices. This confirms a conjecture of Hippchen for $k$-connected graphs when $k\leq 4$ or $k\geq (n-2)/3$.
Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets \(\mathcal{C}\) has the Helly property if any intersecting subcollection has nonempty intersection. A graph is /Helly/ if the collection of maximal complete subgraphs of \(G\) has the Helly property. We prove that if \(G\) is a \(k\)-regular graph with \(n\) vertices such that \(n>3k+\sqrt{2k^{2}-k}\), then the complement \(\bar{G}\) is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equiva…
Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to determine the smallest possible size of the union of the color sets that allows for such a digraph representation. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to introduce the directed intersection number (DIN), …
A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.
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…
On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；National Taiwan University（台湾大学）；VNU University of Science, Hanoi, Vietnam（越南河内国立大学科学大学）
When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ …
