69 research outputs found
A new neighborhood condition for graphs to be fractional (k,m)-deleted graphs
AbstractLet G be a graph of order n, and let k≥2 and m≥0 be two integers. Let h:E(G)→[0,1] be a function. If ∑e∋xh(e)=k holds for each x∈V(G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G):h(e)>0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e)=0 for any e∈E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if δ(G)≥k+2m, n≥8k2+4k−8+8m(k+1)+4m−2k+m−1 and ∣NG(x)∪NG(y)∣≥n2 for any two nonadjacent vertices x and y of G such that NG(x)∩NG(y)≠0̸. Furthermore, it is shown that the result in this paper is best possible in some sense
Factors and Connected Factors in Tough Graphs with High Isolated Toughness
In this paper, we show that every -tough graph with order and isolated
toughness at least has a factor whose degrees are , except for at most
one vertex with degree . Using this result, we conclude that every
-tough graph with order and isolated toughness at least has a
connected factor whose degrees lie in the set , where .
Also, we show that this factor can be found -tree-connected, when is a
-tough graph with order and isolated toughness at least ,
where and . Next, we prove that
every -tough graph of order at least with high enough
isolated toughness admits an -tree-connected factor with maximum degree at
most . From this result, we derive that every -tough graph
of order at least three with high enough isolated toughness has a spanning
Eulerian subgraph whose degrees lie in the set . In addition, we
provide a family of -tough graphs with high enough isolated toughness
having no connected even factors with bounded maximum degree
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
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
Uniform density in matroids, matrices and graphs
We give new characterizations for the class of uniformly dense matroids, and
we describe applications to graphic and real representable matroids. We show
that a matroid is uniformly dense if and only if its base polytope contains a
point with constant coordinates, and if and only if there exists a measure on
the bases such that every element of the ground set has equal probability to be
in a random basis with respect to this measure. As one application, we derive
new spectral, structural and classification results for uniformly dense graphic
matroids. In particular, we show that connected regular uniformly dense graphs
are -tough and thus contain a (near-)perfect matching. As a second
application, we show that strictly uniformly dense real representable matroids
can be represented by projection matrices with constant diagonal and that they
are parametrized by a subvariety of the real Grassmannian.Comment: 23 page
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