Cycles and paths through specified vertices in k-connected graphs

Egawa, Yoshimi; Glas, R.; Locke, Stephen C.

doi:10.1016/0095-8956(91)90086-y

Cited by 31 publications

(30 citation statements)

References 6 publications

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“…Note that other results on cycles through subsets of vertices were discovered by Egawa et al [104]. Broersma et al [79] extended Theorem 3.51 by Bauer et al as follows.…”

Section: This Theorem Immediately Implies the Following Corollarymentioning

confidence: 50%

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Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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“…Note that other results on cycles through subsets of vertices were discovered by Egawa et al [104]. Broersma et al [79] extended Theorem 3.51 by Bauer et al as follows.…”

Section: This Theorem Immediately Implies the Following Corollarymentioning

confidence: 50%

“…Recently Häggkvist and Mader [157] showed that every set of k + ⌊ 1 3 √ k⌋ vertices in a k-connected k-regular graph belongs to some cycle. Egawa et al [104] proved the following common generalizations of Theorem 5.1 and Dirac's classical theorem [101] on hamiltonicity.…”

Section: Cycles Through Specified Verticesmentioning

confidence: 99%

Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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“…Some stronger results have been given for specific classes of graphs, like 3-connected cubic graphs [EHL84,EGL91]. For this class of graphs it is known that there exists a cycle through any 9 vertices, and that there exists a cycle which passes through any 10 given vertices if and only if the graph is not contractible to the Petersen graph [EHL84] in such a way that each of the 10 vertices maps to a distinct vertex of the Petersen graph.…”

Section: Background and Motivationmentioning

confidence: 99%

“…For instance, it is well known that in a k-vertex-connected graph any subset of k nodes [Dir60] or any subset of k − 1 independent edges [HT82] is included in a cycle. There are a number of works giving necessary or sufficient conditions for the existence of a cycle through a specified set of vertices in a general graph [Kaw04,BL81,EGL91,KL82].…”

Section: Background and Motivationmentioning

confidence: 99%

Edge-Simple Circuits through 10 Ordered Vertices in Square Grids

Coudert

Giroire

2009

Lecture Notes in Computer Science

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A circuit in a simple undirected graph G = (V, E) is a sequence of vertices {v 1 , v 2 , . . . , v k+1 } such that v 1 = v k+1 and {v i , v i+i } ∈ E for i = 1, . . . , k. A circuit C is said to be edge-simple if no edge of G is used twice in C. In this article we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of an infinite square grid, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We prove that k = 10. To this end, we first provide a counterexample implying that k < 11. To show that k ≥ 10, we introduce a methodology, based on the notion of core graph, to reduce drastically the number of possible vertex configurations, and then we test each one of the resulting configurations with an ILP solver.

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“…In the literature the question has been studied whether for a given graph G any subset S of vertices of restricted size has some cycle passing through it. Many results on general graphs and graph classes are known (see, e.g., [2], [4], [5] [7], [8], [10], [12], [14], [15], [17]). For triangle-free graphs the following result has been proven.…”

Section: Supported By Jsps Kakenhi (14740087)mentioning

confidence: 99%