A simple proof of a theorem of Jung

Bauer, Douglas C.; Morgana, A.; Schmeichel, E. F.

doi:10.1016/0012-365x(90)90029-h

Cited by 6 publications

(11 citation statements)

References 4 publications

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“…By (3) and 4 In particular, every vertex of V(C) -(A U {y,, y:}) is an i-vertex for some ie{l,... But then o(G -(A U {y,})) 3 IA U {y,} 1 + 1, our final contradiction.…”

Section: Case 21 X' = Y;mentioning

confidence: 82%

“…Theorem 9 allows us to draw conclusions concerning long, but not necessarily hamiltonian, cycles in G. However if 6 = (Y -1. 3 > 'n we cannot conclude from Theorem 9 that G is hamiltonian. It is possible, however, to combine Lemma 8 with a suitably modified proof of Theorem 13 to obtain the following.…”

Section: Theorem W Let G Be a L-tough Graph On N 2 3 Vertices With 6mentioning

confidence: 93%

“…By applying Theorem 5 instead of Theorem 7 it is possible to obtain a new proof of Jung's entire theorem (n 2 11). Details will appear elsewhere [2,3].…”

Section: Conjecturesmentioning

confidence: 99%

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Long cycles in graphs with large degree sums

Bauer¹,

Veldman

Morgana

et al. 1990

Discrete Mathematics

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A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d(x) + d(y) + d(z) 3s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and (Y the cardinality of a maximum independent set of vertices. If G is l-tough and s an, then every longest cycle in G is a dominating cycle and c z min(n, n + fs-cu) >, min(n , $n + 4s) 3 &a. If G is 2-connected and s 2 n + 2, then also c 3 min(n, n + 4s-(u), generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s 2 n, then G is hamiltonian.

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“…By (3) and 4 In particular, every vertex of V(C) -(A U {y,, y:}) is an i-vertex for some ie{l,... But then o(G -(A U {y,})) 3 IA U {y,} 1 + 1, our final contradiction.…”

Section: Case 21 X' = Y;mentioning

confidence: 82%

Section: Theorem W Let G Be a L-tough Graph On N 2 3 Vertices With 6mentioning

confidence: 93%

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Long cycles in graphs with large degree sums

Bauer¹,

Veldman

Morgana

et al. 1990

Discrete Mathematics

Self Cite

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“…As mentioned earlier, our constructive proof of Theorem B shows that within the class of graphs with 2 ¿ n − 4, the properties of being 1-tough and of having a Hamilton cycle can be recognized in polynomial time. Our proof is based on the proof of Jung's Theorem in [2] and the proof of Theorem D in [4]. At the time these results were established, the computational complexity of recognizing 1-tough graphs was not known.…”

Section: Discussionmentioning

confidence: 99%

“…In Section 3, we ÿrst brie y discuss the constructive proof of Theorem A in [7], and then provide a detailed constructive proof of Theorem B (for n ¿ 16). This later proof makes use of arguments that appear in [2,4].…”

Section: Produce a Longer Cycle;mentioning

confidence: 99%

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

Bauer

Broersma

Morgana

et al. 2002

Discrete Applied Mathematics

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A number of results in hamiltonian graph theory are of the form "P1 implies P2", where P1 is a property of graphs that is NP-hard and P2 is a cycle structure property of graphs that is also NP-hard. An example of such a theorem is the well-known ChvÃ atal-Erdős Theorem, which states that every graph G with 6 Ä is hamiltonian. Here Ä is the vertex connectivity of G and is the cardinality of a largest set of independent vertices of G. In another paper ChvÃ atal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than Ä independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.

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Toughness in Graphs – A Survey

Bauer

Broersma

Schmeichel

2006

Graphs and Combinatorics

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107

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In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

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A simple proof of a theorem of Jung

Cited by 6 publications

References 4 publications

Long cycles in graphs with large degree sums

Long cycles in graphs with large degree sums

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

Toughness in Graphs – A Survey

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