Sufficient conditions for hamiltonian properties of graphs

Abstract

We study graph theory. The graphs we consider in the thesis consist of a finite vertex set and a finite edge set, and each edge joins an unordered pair of (not necessarily distinct) vertices. In a graph, a round trip which visits a subset of the vertices once, is called a cycle. If such a round trip passes through every vertex of the graph precisely once, we call it a Hamilton cycle. The Hamilton problem, which is the problem of determining whether a Hamilton cycle exists in a given graph has been proved to be NP-complete. So most researchers are focussing on establishing sufficient conditions, i.e., conditions on the graph that guarantee the existence of a Hamilton cycle. In this thesis, we focus on giving sufficient conditions for hamiltonian properties of a graph in terms of structural or algebraic parameters. Besides the Hamilton cycle, we also consider traceability. It requires that there exists a Hamilton path in the graph, i.e., a path through the vertices and edges of the graph that visits all the vertices exactly once (and does not return to the starting vertex). We say a graph is traceable if it contains a Hamilton path, and traceable from a vertex x if it contains a Hamilton x-path, i.e., a Hamilton path starting at vertex x. So, the next hamiltonian property we deal with is that the graph is traceable from every arbitrary vertex. The last hamiltonian property we take into account is Hamilton-connectivity, which requires that any two distinct vertices can be connected by a Hamilton path. The results in this thesis mainly involve degree conditions, spectral conditions, and Wiener index and Harary index conditions for traceability, hamiltonicity or Hamilton-connectivity of graphs. All of our results involve the characterization of the exceptional graphs, i.e., all the nonhamiltonian graphs that satisfy the condition. Our contributions extend existing results.