69 research outputs found

    A new neighborhood condition for graphs to be fractional (k,m)-deleted graphs

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    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

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    In this paper, we show that every 11-tough graph with order and isolated toughness at least r+1r+1 has a factor whose degrees are rr, except for at most one vertex with degree r+1r+1. Using this result, we conclude that every 33-tough graph with order and isolated toughness at least r+1r+1 has a connected factor whose degrees lie in the set {r,r+1}\{r,r+1\}, where r≥3r\ge 3. Also, we show that this factor can be found mm-tree-connected, when GG is a (2m+ϵ)(2m+\epsilon)-tough graph with order and isolated toughness at least r+1r+1, where r≥(2m−1)(2m/ϵ+1)r\ge (2m-1)(2m/\epsilon+1) and ϵ>0\epsilon > 0. Next, we prove that every (m+ϵ)(m+\epsilon)-tough graph of order at least 2m2m with high enough isolated toughness admits an mm-tree-connected factor with maximum degree at most 2m+12m+1. From this result, we derive that every (2+ϵ)(2+\epsilon)-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set {2,4}\{2,4\}. In addition, we provide a family of 5/35/3-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

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    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

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    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

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    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 11-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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