212 research outputs found
Forbidden triples and traceability: a characterization
AbstractGiven a connected graph G, a family F of connected graphs is called a forbidden family if no induced subgraph of G is isomorphic to any graph in F. If this is the case, G is said to be F-free. In earlier papers the authors identified four distinct families of triples of subgraphs that imply traceability when they are forbidden in sufficiently large graphs. In this paper the authors introduce a fifth family and show these are all such families
Arbitrarily traceable graphs and digraphs
AbstractThe work in this paper extends and generalizes earlier work by Ore on arbitrarily traceable Euler graphs, by Harary on arbitrarily traceable digraphs, by Chartrand and White on randomly n-traversable graphs, and by Chartrand and Lick on randomly Eulerian digraphs. Arbitrarily traceable graphs of mixed type are defined and characterized in terms of a class of forbidden graphs. Arbitrarily traceable digraphs of mixed type are also defined and a simply applied characterization is given for them
On hamiltonicity of 1-tough triangle-free graphs
Let Ï(G) denote the number of components of a graph G. A connected graph G is said to be 1-tough if Ï(Gâ
ââ
X)â€|X| for all XâââV(G) with Ï(Gâ
ââ
X)>1. It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in general, and even not for triangle-free graphs. We present two classes of triangle-free graphs for which the reverse statement holds, i.e., for which hamiltonicity and 1-toughness are equivalent. Our two main results give partial answers to two conjectures due to Nikoghosyan.</p
Global cycle properties in graphs with large minimum clustering coefficient
The clustering coefficient of a vertex in a graph is the proportion of
neighbours of the vertex that are adjacent. The minimum clustering coefficient
of a graph is the smallest clustering coefficient taken over all vertices. A
complete structural characterization of those locally connected graphs, with
minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully
cycle extendable is given in terms of strongly induced subgraphs with given
attachment sets. Moreover, it is shown that all locally connected graphs with
minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly
pancyclic, thereby proving Ryjacek's conjecture for this class of locally
connected graphs.Comment: 16 pages, two figure
Spanning Halin Subgraphs Involving Forbidden Subgraphs
In structural graph theory, connectivity is an important notation with a lot of applications. Tutte, in 1961, showed that a simple graph is 3-connected if and only if it can be generated from a wheel graph by repeatedly adding edges between nonadjacent vertices and applying vertex splitting. In 1971, Halin constructed a class of edge-minimal 3-connected planar graphs, which are a generalization of wheel graphs and later were named âHalin graphsâ by Lovasz and Plummer. A Halin graph is obtained from a plane embedding of a tree with no stems having degree 2 by adding a cycle through its leaves in the natural order determined according to the embedding. Since Halin graphs were introduced, many useful properties, such as Hamiltonian, hamiltonian-connected and pancyclic, have been discovered. Hence, it will reveal many properties of a graph if we know the graph contains a spanning Halin subgraph. But unfortunately, until now, there is no positive result showing under which conditions a graph contains a spanning Halin subgraph. In this thesis, we characterize all forbidden pairs implying graphs containing spanning Halin subgraphs. Consequently, we provide a complete proof conjecture of Chen et al. Our proofs are based on Chudnovsky and Seymourâs decomposition theorem of claw-free graphs, which were published recently in a series of papers
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
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