163 research outputs found
Route systems of a connected graph
summary:The concept of a route system was introduced by the present author in [3].Route systems of a connected graph generalize the set of all shortest paths in . In this paper some properties of route systems are studied
The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs
summary:If is a connected graph of order , then by a hamiltonian coloring of we mean a mapping of into the set of all positive integers such that (where denotes the length of a longest path in ) for all distinct . Let be a connected graph. By the hamiltonian chromatic number of we mean where the minimum is taken over all hamiltonian colorings of . The main result of this paper can be formulated as follows: Let be a connected graph of order . Assume that there exists a subgraph of such that is a hamiltonian-connected graph of order , where . Then
The interval function of a connected graph and a characterization of geodetic graphs
summary:The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties of the interval function
A characterization of the interval function of a (finite or infinite) connected graph
summary:By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph
- …