489 research outputs found
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
Extremal and degree donditions for path extendability in digraphs
In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles
On the structure of the directions not determined by a large affine point set
Given a point set in an -dimensional affine space of size
, we obtain information on the structure of the set of
directions that are not determined by , and we describe an application in
the theory of partial ovoids of certain partial geometries
Stallings graphs for quasi-convex subgroups
We show that one can define and effectively compute Stallings graphs for
quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or
right-angled Artin groups). These Stallings graphs are finite labeled graphs,
which are canonically associated with the corresponding subgroups. We show that
this notion of Stallings graphs allows a unified approach to many algorithmic
problems: some which had already been solved like the generalized membership
problem or the computation of a quasi-convexity constant (Kapovich, 1996); and
others such as the computation of intersections, the conjugacy or the almost
malnormality problems.
Our results extend earlier algorithmic results for the more restricted class
of virtually free groups. We also extend our construction to relatively
quasi-convex subgroups of relatively hyperbolic groups, under certain
additional conditions.Comment: 40 pages. New and improved versio
Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs
The study of cycles, particularly Hamiltonian cycles, is very important in many applications.
Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity.
An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable.
In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable.
Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented.
The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable.
The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case.
Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
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