65 research outputs found

    Toughness and hamiltonicity in kk-trees

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    We consider toughness conditions that guarantee the existence of a hamiltonian cycle in kk-trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to 74\frac{7}{4}. It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to kk-trees for k≥2k\ge 2: Let GG be a kk-tree. If GG has toughness at least k+13,\frac{k+1}{3}, then GG is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough kk-trees for each $k\ge 3

    On some intriguing problems in Hamiltonian graph theory -- A survey

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    We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, tt-tough graphs, and claw-free graphs

    How tough is toughness?

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    The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors

    Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.

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    We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time

    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

    Forbidden subgraphs for Hamiltonian problems on 2-trees

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    The Hamiltonian path problem is a well-known NP-complete graph theory problem which is to determine whether or not it is possible to find a spanning path in a graph. Some variations on this problem include the 1HP and 2HP problems, which are to determine whether or not it is possible to find a Hamiltonian path in a graph if one or two endpoints of the path are fixed, respectively. Both problems are also NP-complete for graphs in general, though like the Hamiltonian path problem, they are polynomially solvable on certain types of graphs. 2-trees are a specific type of graph for which the 1HP, 2HP, and traditional Hamiltonian path problems are polynomially solvable. It is known that 2-trees have a Hamiltonian cycle if and only if they are 1-tough. However, the analogous statement for Hamiltonian paths does not hold. We will structurally characterize 2HP on 2-trees, and then use these results to structurally characterize 1HP and HP on 2-trees. We will define a family of 2-trees such that any 2-tree has a Hamiltonian path if and only if it does not contain any graph from that family as an induced graph
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