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Improved Sufficient Conditions for Hamiltonian Properties

100%
Bode J.-P., Fricke A., Kemnitz A.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 2
329-334
EN
In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition.
2
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On a conjecture of quintas and arc-traceability in upset tournaments

100%
Busch A., Jacobson M., Reid K.
Discussiones Mathematicae Graph Theory
|
2005
|
tom 25
|
nr 3
225-239
EN
A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of an upset tournament lie on a hamiltonian path, and deduce a characterization of arc-traceable upset tournaments.
3
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Every 8-Traceable Oriented Graph Is Traceable

88%
Aardt S. A. v.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 4
963-973
EN
A digraph of order n is k-traceable if n ≥ k and each of its induced subdigraphs of order k is traceable. It is known that if 2 ≤ k ≤ 6, every k-traceable oriented graph is traceable but for k = 7 and for each k ≥ 9, there exist k-traceable oriented graphs that are nontraceable. We show that every 8-traceable oriented graph is traceable.
4
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Decompositions into two paths

88%
Skupień Z.
Discussiones Mathematicae Graph Theory
|
2005
|
tom 25
|
nr 3
325-329
EN
It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.
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