This paper presents a classification of vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. The classification leads to constructions of new examples of half-arc-transitive graphs.

A Hamiltonian path is a spanning path in a graph i.e. a path through every vertex. In this paper we present an interesting sufficient condition for a graph to possess a Hamiltonian path. In particular we prove that the degree sum of all pairwise nonadjacent vertex-triples is greater than 1/2(3n 5) implies that the graph has a Hamiltonian path, where n is the number of vertices of that graph. Al...

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in t...

We propose solving computational problems with DNA molecules by physically constructing 3-dimensional graph structures. Building blocks consisting of intertwined strands of DNA are used to represent graph edges and vertices. Diierent blocks would be combined to form all possible 3-dimensional structures representing a graph. The solution to the Hamiltonian cycle problem provided requires a cons...

a r t i c l e i n f o a b s t r a c t Keywords: Graph algorithms Simple-path convexity Simple-path convex hull Totally balanced hypergraphs In a connected hypergraph a vertex set X is simple-path convex (sp-convex, for short) if either |X| 1 or X contains every vertex on every simple path between two vertices in X (Faber and Jamison, 1986 [7]), and the sp-convex hull of a vertex set X is the mi...

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any n-vertex path-pairable planar graph must contain a vertex of degree linear in n.

Two new path problems in graphs are studied: MINRANGE, i.e., find a path from a vertex s to a vertex 1 with the smallest possible range of arc lengths. and MINRATIO, i.e., find such a path for which the ratio of the largest to the smallest arc length is minimum. Several bicriterion extensions of these problems are also considered. Ke_vwords: Bicriterion paths; Range; Ratio

The path-table P(T ) of a tree T collects information regarding the paths in T: for each vertex v, the row of P(T ) relative to v lists the number of paths containing v of the various lengths. We call this row the path-row of v in T. Two trees having the same path-table (up to reordering the rows) are called path-congruent (or path-isomorphic). Motivated byKelly–Ulam’sReconstructionConjecture a...

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the comp...

We study the problem of finding a short path from a start to a goal within a twodimensional continuous and isotropic terrain that has been discretized into an array of accessible and blocked cells. A classic approach obtains a grid path where each step is along the edge of an accessible cell or diagonally across one. Grid paths suffer from ‘digitization bias’ – even if two locations have line-o...