No abstract

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 2free 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.

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, respectively. In this paper, we show that every 15-tough $(P_2\cup P_3)$-free graph with at least three vertices is hamiltonian.

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, we confirm the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST, which was proposed as a question by Albertson, Berman, Hutchinson and Thomassen.

An antimagic labeling of a directed graph D with n vertices and m arcs is a bijection from the set of arcs of D to the integers {1, · · · , m} such that all n oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph G is said to have an antimagic orientation if G has an orientation which admits an antimagic labeling. Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every biregular bipartite graph admits an antimagic orientation. toward this conjecture. Let G be a graph with n vertices other than K 2 . In 2004, Alon, Kaplan, Lev, Roditty, and Yuster [1] showed that there exists a constant c such that if G has minimum degree at least c · logn, then G is antimagic. They also proved that G is antimagic when the maximum degree of G is at least n − 2, and they proved that all complete multipartite graphs (other than K 2 ) are antimagic. The latter result of Alon et al.was improved by Yilma [14] in 2013.Apart from the above results on dense graphs, the antimagic labeling conjecture has been also verified for regular graphs. Started with Cranston [5] showing that every bipartite regular graph is antimagic, regular graphs of odd degree [6], and finally all regular graphs [3] were shown to be antimatic sequentially. For more results on the antimagic labeling conjecture for other classes of graphs, see [7,9,11,12].Hefetz, Mütze, and Schwartz [10] introduced the variation of antimagic labelings, i.e., antimagic labelings on directed graphs. An antimagic labeling of a directed graph with m arcs is a bijection from the set of arcs to the integers {1, ..., m} such that any two oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A digraph is called antimagic if it admits an antimagic labeling. For an undirected graph G, if it has an orientation such that the orientation is antimagic, then we say G admits an antimagic orientation. Hefetz et al. in the same paper posted the following problems.Question 1 ([10]). Is every connected directed graph with at least 4 vertices antimagic?Conjecture 1.1 ([10]). Every connected graph admits an antimagic orientation.Parallel to the results the on antimagic labelling conjecture, Hefetz, Mütze, and Schwartz [10] showed that every orientation of a dense graph is antimagic and almost all regular graphs have an antimagic orientation. Particulary, they showed that every orientation of stars (other than K 1,2 ), wheels, and complete graphs (other than K 3 ) is antimagic. Observe that if a bipartite graph is antimagic, then it has an antimagic orientation obtained by directing all edges from one partite set to the other. Thus by the result of Cranston [5], regular bipartite graphs have an antimagic orientation. A bipartite g...