Nontraceable detour graphs

AbstractThe detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore detour graphs with detour deficiency zero. In this paper, we give a number of constructions for detour graphs of all orders greater than 17 and all detour deficiencies greater than zero. These constructions are used to give examples of nontraceable detour graphs with chromatic number k, k⩾2, and girths up to 7. Moreover we show that, for all positive integers l⩾1 and k⩾3, there are nontraceable detour graphs with chromatic number k and detour deficiency l

Maximal nontraceable graphs

A graph G is maximal nontraceable (MNT) (maximal nonhamiltonian (MNH)) if G is not traceable (hamiltonian), i.e. does not contain a hamiltonian path (cycle), but G+xy is traceable (hamiltonian) for all nonadjacent vertices x and y in G. A graph G is hypohamiltonian if G is not hamiltonian, but every vertex deleted subgraph G -u of G is hamiltonian. A graph which is maximal nonhamiltonian and hypohamiltonian is called maximal hypohamiltonian (MHH). Until recently, not much has appeared in the literature about MNT graphs, although there is an extensive literature on MNH graphs. In 1998 Zelinka constructed two classes of MNT graphs and made the conjecture, which he later retracted, that every MNT graph belongs to one of these classes. We show that there are many different types of MNT graphs that cannot be constructed by Zelinka's methods. Although we have not been able to characterize MNT graphs in general, our attempt at characterizing MNT graphs with a specified number of blocks and cut-vertices enabled us to construct infinite families of non-Zelinka MNT graphs which have either two or three blocks. We consider MNT graphs with toughness less than one, obtaining results leading to interesting constructions of MNT graphs, some based on MHH graphs. One result led us to discover a non-Zelinka MNT graph of smallest order, namely of order 8. We also present examples of MNTgraphs with toughness at least one, including an infinite family of 2-connected, claw-free graphs. We find a lower bound for the size of 2-connected MNT graphs of order n. We construct an infinite family of 2-connected cubic MNT graphs of order n, using MHH graphs as building blocks. We thus find the minimum size of 2-connected MNT graphs for infinitely many values of n. We also present a construction, based on MHH graphs, of an infinite family of MNT graphs that are almost cubic. We establish the minimum size of MNT graphs of order n, for all except 26 values of n, and we present a table of MNT graphs of possible smallest size for the excluded 26 values of n.Mathematical SciencesPHD (MATHEMATICS

Local properties of graphs

We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated.Mathematical SciencesD. Phil. (Mathematics

On the path sequence of a graph

A subset S of vertices of a graph G = (V;E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by k(G) the minimum cardinality of a k-path vertex cover in G and form a sequence (G) = ( 1(G); 2(G); : : : ; jV j(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in (G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

On the Path Sequence of a Graph

A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by Ψk (G) the minimum cardinality of a k-path vertex cover in G and form a sequence Ψ (G) = (Ψ1 (G), Ψ2 (G), . . . , Ψ|V| (G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in Ψ(G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

The Ticker, April 1, 1998

The Ticker is the student newspaper of Baruch College. It has been published continuously since 1932, when the Baruch College campus was the School of Business and Civic Administration of the City College of New York