Independent sets which meet all longest paths

Galeana‐Sánchez, Hortensia; Rincón-Mejía, Hugo Alberto

doi:10.1016/0012-365x(94)00261-g

Cited by 7 publications

(5 citation statements)

References 4 publications

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“…[9,1]) but the conjecture seems very difficult to settle in general. Even the special case a = 1, which was formulated by Laborde, Payan, and Xuong in 1982 and treated in [9,4,5,7,8], has not yet been settled. Another interesting special case that remains open is the restriction of the DPPC to 1-deficient oriented graphs (see [10,12]).…”

Section: Conjecture 2 (Dppc) If D Is a Digraph And (A B) Is Any Paimentioning

confidence: 98%

Traceability of k-traceable oriented graphs

Aardt

Dunbar

Frick

et al. 2010

Discrete Mathematics

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a b s t r a c tA digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable.We note that 2-traceable oriented graphs are tournaments and for k ≥ 3, k-traceable oriented graphs can be regarded as generalized tournaments. We show that for 2 ≤ k ≤ 6 every k-traceable oriented graph is traceable, thus extending the well-known fact that every tournament is traceable. This result does not extend to k = 7. In fact, for every k ≥ 7, except possibly for k = 8 or 10, there exist k-traceable oriented graphs that are nontraceable. However, we show that for every k ≥ 2 there exists a smallest integer t(k) such that every k-traceable oriented graph of order at least t(k) is traceable.

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Section: Conjecture 2 (Dppc) If D Is a Digraph And (A B) Is Any Paimentioning

confidence: 98%

Traceability of k-traceable oriented graphs

Aardt

Dunbar

Frick

et al. 2010

Discrete Mathematics

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“…In 1995 Bondy [13] stated a conjecture that is close to the digraph analogue of the PPC, but the PPC for undirected graphs was not mentioned in the literature again until 1997, when it was addressed in Problem 1 of [15] and also in [20]. Since then, several authors have contributed results towards proving the PPC (see [17,18,21,22,25,26,29,30,33,34,45,46,47]) and its digraph analogue, the DPPC (see [1,2,3,4,5,6,8,9,28,31,32,37]). The PPC and DPPC are also discussed in the text books [35] and [10], respectively.…”

Section: Frickmentioning

confidence: 99%

“…This special case has turned out to be a difficult and intriguing conjecture in its own right. It is treated, for example, in [2,31,32,37,42].…”

Section: Generalized Chromatic Numbersmentioning

confidence: 99%

A survey of the Path Partition Conjecture

Frick¹

2013

Discuss. Math. Graph Theory

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The Path Partition Conjecture (PPC) states that if G is any graph and (λ 1 , λ 2 ) any pair of positive integers such that G has no path with more than λ 1 + λ 2 vertices, then there exists a partition (V 1 , V 2 ) of the vertex set of G such that V i has no path with more than λ i vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the PPC that have appeared in the literature since its first formulation in 1981.

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“…The TC is, in turn, motivated by the celebrated Path Partition Conjecture for Digraphs (DPPC) [15] dating back to 1982, which has been proven correct for various special cases [8,2], but seems to be very difficult to settle in general -even the case a = 1 has not yet been settled [10,11,14].…”

Section: Introductionmentioning

confidence: 99%

Computational Results on the Traceability of Oriented Graphs of Small Order

Burger

2013

Electron. J. Combin.

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A digraph $D$ is traceable if it contains a path visiting every vertex, and hypotraceable if $D$ is not traceable, but $D-v$ is traceable for every vertex $v\in V(D)$. Van Aardt, Frick, Katrenič and Nielsen [Discrete Math. 11(2011), 1273-1280] showed that there exists a hypotraceable oriented graph of order $n$ for every $n\geq 8$, except possibly for $n=9$ or $11$. These two outstanding existence questions for hypotraceable oriented graphs are settled in this paper --- the first in the negative and the second in the affirmative. Furthermore, $D$ is $k$-traceable if $D$ has at least $k$ vertices and each of its induced subdigraphs of order $k$ is traceable. It is known that for $k\leq 6$ every k-traceable oriented graph is traceable and that for $k=7$ and each $k\geq 9$ there exist nontraceable $k$-traceable oriented graphs of order $k+1$. The Traceability Conjecture states that for $k\geq 2$ every $k$-traceable oriented graph of order $n\geq 2k-1$ is traceable. In this paper it is shown via computer searches that all $7$-traceable and $8$-traceable oriented graphs of orders $9$, $10$ and $11$ are traceable, and that all $9$-traceable oriented graphs of order $11$ are traceable. All hypotraceable graphs of order $10$ are also found. Recently, these results are used to prove that the Traceability Conjecture also holds for $k =7, 8$ and 9, except possibly when $k=9$ and $22\leq n\leq 32$.

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Independent sets which meet all longest paths

Cited by 7 publications

References 4 publications

Traceability of k-traceable oriented graphs

Traceability of k-traceable oriented graphs

A survey of the Path Partition Conjecture

Computational Results on the Traceability of Oriented Graphs of Small Order

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