H. Fleischner. The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory B, 16:29-34, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 2 and let z 1 ; z 2 be endvertices of two different leaves of F . Then, deleting from F the edges xz 1 , xz 2 and adding z 1 z 2 , we get a flower F 0 with d F 0 (x) d F (x) which contradicts the minimality of F . Thus, d F (x) 2 and F is a hamiltonian cycle. 3. SQUARES Fleischner [2] proved the following theorem. Theorem A. 2] If H is a 2 connected graph and G = H 2 , then G is hamiltonian. The following statement is also due to Fleischner and follows from Theorem 3 of [3] Theorem B. 3] Let y be an arbitrary vertex of a 2 connected graph H. Then the graph G = H 2 ....

....different leaves of F . Then, deleting from F the edges xz 1 , xz 2 and adding z 1 z 2 , we get a flower F 0 with d F 0 (x) d F (x) which contradicts the minimality of F . Thus, d F (x) 2 and F is a hamiltonian cycle. 3. SQUARES Fleischner [2] proved the following theorem. Theorem A. [2] If H is a 2 connected graph and G = H 2 , then G is hamiltonian. The following statement is also due to Fleischner and follows from Theorem 3 of [3] Theorem B. 3] Let y be an arbitrary vertex of a 2 connected graph H. Then the graph G = H 2 contains a hamiltonian cycle C such that both ....

Fleischner, H.: The square of every two-connected graph is hamiltonian. J. Combin. Theory Ser. B 16(1974), 29-34.

....by structure or by density) although such results abound [1] 9] 38] 41] 42] Similarly, excepting Theorem 2. 1, which indicates how close general graphs come to being pancyclic, we do not discuss results concerning smalldilation embeddings of cycles in graphs, although such results exist [16]. Finally, we do not discuss the computational difficulty of detecting various classes of cycles in various classes of host graphs. We do remark, however, that the general problem of deciding the containment of cycles in graphs is as computationally intractable as the general question of deciding ....

H. Fleischner (1974): The square of every two-connected graph is hamiltonian. J. Comb. Th. (B) 16, 29-34. 33

....a 4 approximation. The 8 approximation algorithm is described below. 1. Find an approximation to the minimum cost 2 node connected subgraph of G. Call this graph S. 2. Find a Hamiltonian cycle in S 2 . In 1970, Fleischner proved that the square of a 2 node connected graph is Hamiltonian [6, 7], and Lau, in his Ph.D. thesis, provided a constructive proof [9, 10] that finds a Hamiltonian cycle in O(n 2 ) time. We note that Fleischner s result is applied to the tsp in a similar way by Parker and Rardin [14] who show a 2 approximation for the bottleneck tsp. Lemma 1 Any Hamiltonian ....

H. Fleischner. The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory, 16:29--34, 1974.

....= AO(G) M e , so that AO(G) M e is also (n Gamma c) connected. In addition, AO(G) is bipartite and, by Lemma 1.1, has minimum degree at least n Gamma c. Thus, Theorem 2.3 applies and therefore establishes that AO(G) is (n Gamma c) connected. Although AO(G) is not always Hamiltonian, Fleischner [5, 6, 11] has shown that the square of any 2 connected graph with at least three vertices is Hamiltonian and therefore AO 2 (G) is Hamiltonian when n Gamma c 2. An explicit construction of a Hamiltonian cycle in AO 2 (G) is given in [16] An eulerian spanning subgraph of a graph G is a closed trail ....

H. Fleischner, The square of every two-connected graph is Hamiltonian, J. Combin. Theory Ser. B, 16 (1974) 29-34.

....u and v using at most 2 edges. Our algorithm is described below. tsp approx (G) 1. Find an (approximately) minimum cost 2 node connected subgraph of G. Call this graph S. 2. Find a Hamiltonian cycle in S 2 . In 1970, Fleischner proved that the square of a 2 node connected graph is Hamiltonian [7, 8], and in his Ph.D. thesis Lau provided a constructive proof [11, 12] that finds a Hamiltonian cycle in polynomial time. We note that Fleischner s result is applied to the tsp in a simpler way by Parker and Rardin [17] who show a 2 approximation for the bottleneck tsp. Theorem 1 Algorithm ....

H. Fleischner. The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory, 16:29--34, 1974.

....[19] is that the cube of every connected graph is Hamiltonian. Since G 0 (P) is always connected, G 0 (P) 3 is Hamiltonian and a delay 3 ordering exists. The graph G 0 (P) is not always 2 connected; otherwise the existence of a delay 2 ordering would by implied by a result of Fleischner [7] which states that the square of every 2 connected graph is Hamiltonian. Even though G 0 (P) is not in general 2 connected, the posets with 2 connected transposition graphs are easy to characterize. First, consider the question of which transposition graphs have pendant vertices. If P consists ....

H. Fleischner, The square of every two-connected graph is Hamiltonian, J. Combin. Theory (B), 16 (1974), pp. 29--34.

....problem in general. Furthermore, note that the characterization given by Mukhopadhyay dealing with some n cliques of G which will have exponentially many candidate cliques. Indeed, Motwani and Sudan [20] showed that recognizing the general square graphs is an NP complete problem. Fleischner [8] proved that the square of a biconnected graph is always Hamiltonian. Although the Hamiltonian cycle problem is NP complete for general graphs [10] the biconnectivity of a graph can be tested in linear time [25] Thus an efficient algorithm for finding the square root of some special classes of ....

....of finding the maximum cliques in powers of graphs by transformations from the general maximum cliques problem. In Section 5.6, we prove the chordality of powers of trees by showing that they have the intersection model of subtrees of a tree. 5. 1 Hamiltonian Cycles in Powers of Graphs Fleischner [8] proved that the square of a biconnected graph is always Hamiltonian. Harary and Schwenk [17] proved that the square of a tree T is Hamiltonian if and only if v 0 u 0 v u T 2 T 1 Figure 6: Finding a Hamiltonian cycle in a cubic tree. T does not contain S(K 1;3 ) as its induced subgraph. Here ....

H. Fleischner. The square of every two-connected graph is Hamiltonian. J. Combin. Theory B, 16:29--34, 1974.

....# in G. Our approximation ratio cannot be polynomially improved in the sense that no (n 4) 1 2 # approximation, for any fixed # 0, can be obtained in polynomial time unless NP = ZPP. Studying NP hard problems in powers of graphs is a topic that has received some attention in the literature [5, 34, 18, 4]. Independently of our work, Baveja and Srinivasan (personal communication) have obtained results similar to ours for approximating vertex disjoint paths under the rounding approach, unsplittable flow and column restricted packing integer programs. Their work builds on the methods in [31] 2 ....

H. Fleischner. The square of every two-connected graph is hamiltonian. J. of Combinatorial Theory B, 16:29--34, 1974.

....We will call G a k th root of the graph G k . In particular, G 2 is the square of the graph G and G is a square root of the graph G 2 . Powers of graphs have been studied extensively in graph theory. For example, it is well known that the square of a 2 connected graph has a Hamiltonian cycle [2], and the Hamiltonian cycle Supported by Mitsubishi Corporation, NSF Grant CCR 9010517 and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. y Part of this research was done while this author was a student ....

H. Fleischner, The square of every two-connected graph is Hamiltonian, Journal of Combinatorial Theory (Series B), 16 (1974), pp. 29--34.

.... are defined in the same way as for undirected graphs by taking paths and cycles as directed ones, and this can also be done regarding the construction of higher powers of G, namely: The k th power G k of G is the digraph G k = V k ; E k ) with V k : V and E k : f(x; y) x; y 2 V 1 dG (x; y) kg; k = 1; 2; where dG (x; y) denotes the distance from x to y in G, that is the length of a shortest path from x to y in G if there is any and 1 otherwise. In spite of these analogous definitions the Hamiltonian properties of powers of undirected and directed graphs are ....

.... powers of G, namely: The k th power G k of G is the digraph G k = V k ; E k ) with V k : V and E k : f(x; y) x; y 2 V 1 dG (x; y) kg; k = 1; 2; where dG (x; y) denotes the distance from x to y in G, that is the length of a shortest path from x to y in G if there is any and 1 otherwise. In spite of these analogous definitions the Hamiltonian properties of powers of undirected and directed graphs are essentially different. A first difference is rather evident: For an undirected connected graph G on n 3 vertices, G 3 is Hamiltonian. For a connected digraph G, ....

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Fleischner, H., The square of every two-connected graph is Hamiltonian, J. Combinatorial Theory 3 (1974), 29-34

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H. Fleischner. The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory B, 16:29-34, 1974.

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H. Fleischner. The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory B, 16:29--34, 1974.

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