R. J. Gould, Updating the hamiltonian problem - a survey, J. Graph Theory 15 (1991), 121-157.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in N , Q, and M are B S = f1; 2; 5g, B 0 T = f7g (so B T = f6g) and B = f1; 2; 5; 6g, respectively. 3 Antipodal Layers The Antipodal Layers Problem was rst posed by the author during the problem session of the 4 th SIAM Conference on Discrete Mathematics, June 1988. It is also noted in [Go] and attributed to Roth as a personal communication. The graph M n;k is vertex and edgetransitive but cannot be written so nicely as a bowtie product. Nonetheless, we prove the following theorem. Theorem 2 Given any 0 there is an integer k( such that, for k k( and c 1 = 1 ln 3 ; c 2 ....

R.J. Gould, Updating the Hamiltonian Problem{A Survey, J. Graph Theory 15 (1991), 121-157.

....if it does not contain an induced subgraph that is isomorphic to C. Some of the results surveyed here are also mentioned in the two former surveys on claw free graphs, one by Flandrin in [65] and the other by Li Mingchu and Liu Zhenhong in [135] and in the survey paper on hamiltonicity by Gould [87]. t t t t XXXXXXXXXX X Figure 1.1: The claw C Families of claw free graphs There are several well known and important families of graphs that are also claw free, so we now recall some of these families. ff) Line graphs If G is a graph, then the line graph of G, usually denoted by L(G) is ....

....problems for arbitrary graphs. Some results of this type have already been proved and are summarized later in section 5. 2. PATHS, CYCLES, HAMILTONICITY AND RELATED PROBLEMS. a) Preliminaries Many of the results that are mentioned in this section are also included in the survey by Gould [87]. If S ae V (G) then by c(G Gamma S) we denote the number of components of G Gamma S. We say that a graph G is t tough if for every subset S ae V (G) with c(G Gamma S) 1 we have jSj tc(G Gamma S) The toughness of G, denoted by (G) is the largest value of t such that G is t tough. ....

Gould, R.J.: Updating the Hamiltonian problem - a survey. J. Graph Theory 15(1991) 121-157

....the existence of C jW j only. 1 Introduction A graph is hamiltonian if it contains a cycle through all its vertices. Such a cycle is frequently called a hamiltonian cycle. The characterization of these graphs is apparently a very hard problem, though various sufficient conditions are known (cf. [5] for a survey) Many such conditions were given in terms of vertex degrees; the following two theorems are probably the best known representatives. Theorem A. Ore, 9] Let G be a graph of order n 3. If d(x) d(y) n for every pair of non adjacent vertices x; y 2 V (G) then G is hamiltonian. ....

R. J. Gould, "Updating the hamiltonian problem - a survey", J. Graph Theory 15 (1991), 121--157.

....is hamiltonian, is NP complete, and that (up to now) there exists no easily verifiable necessary and su#cient condition for the existence of a Hamilton cycle. This fact gave rise to a growing number of conditions that are either necessary or su#cient. We refer to [7] 8] 11] 12] 25] and [34] for more background and general surveys. Before we turn to our three graph classes, we mention a few results that inspired most of today s work, and give some recent developments that cannot be found in the most recent survey [34] 1.1 Early degree conditions and a closure operation Most of ....

....necessary or su#cient. We refer to [7] 8] 11] 12] 25] and [34] for more background and general surveys. Before we turn to our three graph classes, we mention a few results that inspired most of today s work, and give some recent developments that cannot be found in the most recent survey [34]. 1.1 Early degree conditions and a closure operation Most of the su#cient conditions for hamiltonicity are based on the intuitive idea that a Hamilton cycle is likely to exist if all vertices have many neighbors. The earliest degree condition is based on the minimum degree #(G) of the graph G. ....

R.J. Gould, Updating the hamiltonian problem -- a survey. J. Graph Theory 15 (1991) 121--157.

.... problems that are NP complete in the general setting are polynomially solvable for them [2, 3] Since the middle of the 70s the articles have begun to appear that are devoted to study of conditions sufficient for hamiltonicity of a K 1;3 free graph (see Section 3 of the newest survey by Gould [4]) It is the following conjecture extending Thomassen s conjecture for line graphs [5] that became widely known. Conjecture [6] Every 4 connected K 1;3 free graph is Hamiltonian. It is known that the Hamiltonian cycle problem is NP complete for 3 connected cubic planar graphs [7] Plummer and ....

R. J. Gould, "Updating the Hamiltonian problem---a survey," J. Graph Theory, 15, No. 2, 121-- 157 (1991).

.... for which the Cayley digraph is not hamiltonian [Ran48] This is a special case of the more general conjecture of Lov asz that every connected, undirected, vertex transitive graph has a Hamilton path [Lov70] Results on Hamilton cycles are surveyed in [Als81] for vertex transitive graphs and in [Gou91] for general graphs. A survey of Hamilton cycles in Cayley graphs can be found in [WG84] and in the recent update of Curran and Gallian [CG96] We focus here on a few recent questions which arose in the context of Gray codes. Suppose the group G is S n , the symmetric group of permutations of n ....

....Additional information on Gray codes also appears in the survey of Squire [Squ94a] In [Gol93] Goldberg considers generating combinatorial structures for which achieving even polynomial delay is hard. For surveys on related material, see [Als81] for long cycles in vertex transitive graphs, [Gou91] for hamiltonian cycles, WG84] and the recent update [CG96] for Cayley graphs, and [Sed77] for permutations. Acknowledgements I am grateful to Herb Wilf for collecting and sharing such an intriguing array of Gray code problems. His work, as well as his enthusiasm, has been inspiring. I would ....

R. J. Gould. Updating the hamiltonian problem - a survey. Journal of Graph Theory, 15(2):121 --157, 1991.

....of a single generator or its inverse to its immediate predecessor This problem seems very difficult. For surveys of the general topic of Gray codes, for many more examples of such codes in a variety of combinatorial families, and for pointers to recent literature in the subject we suggest [1, 3, 6, 8]. 1.2. About this paper. In this paper, we study a numerical obstruction to being able to list in Gray code order a collection of subsets of f1; ng. We show that this obstruction grows exponentially in n for the collection of g blockfree subsets of f1; ng if and only if g 2. ....

R. J. Gould, Updating the hamiltonian problem - a survey, Journal of Graph Theory 15, No. 2 (1991), 121-157.

....that G Gamma fu 1 ; u 2 ; u i g has at most two components. The maximum cycle length in G is the circumference c(G) If c(G) n, G is said to be Hamiltonian. The characterization of Hamiltonian graphs is apparently a very hard problem, though various sufficient conditions are known (cf. [3] for a survey) Many of these conditions are based on vertex degrees; such as the following result of Bondy. Theorem 1 [1] Let G be a block with vertex degrees d 1 d 2 Delta Delta Delta dn . If d j j; d k k; j 6= k) d j d k c; then G has a cycle of length at least min(c; n) A ....

Gould R. J., Updating the hamiltonian problem --- a survey, J. Graph Theory 15 (1991), 121--157.

....ones. 1 Introduction and Definitions A graph is hamiltonian if it contains a cycle through all its vertices. Such a cycle is frequently called a hamiltonian cycle. The characterization of these graphs is apparently a very hard problem, though various sufficient conditions are known (cf. [8] for a survey) To be able to state our main result we introduce some definitions and notations. The degree of a vertex v 2 V (G) is denoted by dG (v) and the distance of the vertices u; v is denoted by dist G (u; v) Let N k G (v) N =k G (v) denote the set of all vertices x 2 V (G) for ....

R. J. Gould, Updating the hamiltonian problem - a survey, J. Graph Theory 15, (1991), 121-157.

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R. J. Gould, Updating the hamiltonian problem - a survey, J. Graph Theory 15 (1991), 121-157.

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Gould, R., J.: Updating the hamiltonian problem - a survey, J. Graph Theory 15, 1991, 121-157.

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