A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Ozeki, Kenta; Yamashita, Tomoki

doi:10.1007/s00373-008-0802-z

Cited by 9 publications

(3 citation statements)

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“…If σ k+1 ≥ n + κ + (k − 2)(α − 1), then G is Hamiltonian. Theorem 2 was conjectured by Ozeki and Yamashita [15], and has been proven for small integers k: The case k = 2 of Theorem 2 coincides Theorem 1 (v). The cases k = 1 and k = 3 were shown by Fraisse and Jung [8], and by Ozeki and Yamashita [15], respectively.…”

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confidence: 75%

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A Degree Sum Condition on the Order, the Connectivity and the Independence Number for Hamiltonicity

Chiba

Furuya

Ozeki

et al. 2019

Electron. J. Combin.

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In [Graphs Combin. 24 (2008) 469-483.], the third author and the fifth author conjectured that if G is a k-connected graph such that σ k+1 (G) ≥ |V (G)| + κ(G) + (k − 2)(α(G) − 1), then G contains a Hamiltonian cycle, where σ k+1 (G), κ(G) and α(G) are the minimum degree sum of k + 1 independent vertices, the connectivity and the independence number of G, respectively. In * this paper, we settle this conjecture. This is an improvement of the result obtained by Li: If G is a k-connected graph such that σ k+1 (G) ≥ |V (G)| + (k − 1)(α(G) − 1), then G is Hamiltonian. The degree sum condition is best possible.

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confidence: 75%

“…Proof of Theorem 2. The cases k = 1, k = 2 and k = 3 were shown by Fraisse and Jung [8], by Bauer et al [2] and by Ozeki and Yamashita [15], respectively. Therefore, we may assume that k ≥ 4.…”

Section: Proof Of Theoremmentioning

confidence: 92%

A Degree Sum Condition on the Order, the Connectivity and the Independence Number for Hamiltonicity

Chiba

Furuya

Ozeki

et al. 2019

Electron. J. Combin.

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“…It seems that most of the work on cyclability has been done with the intention of eventually using it to prove that certain graphs are Hamiltonian or because it can be viewed as a relaxation of Hamiltonicity. An interesting list of theorems on cyclability can be found in [21]. Here, we explore cyclability as a tool to obtain approximate TSP tours.…”

Section: Steiner Cyclesmentioning

confidence: 99%

Graph-TSP from Steiner Cycles

Iwata

Newman

Ravi

2014

Graph-Theoretic Concepts in Computer Science

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Abstract. We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph G, if we can find a spanning tree T and a simple cycle that contains the vertices with odd-degree in T , then we show how to combine the classic "double spanning tree" algorithm with Christofides' algorithm to obtain a TSP tour of length at most . We use this approach to show that a graph containing a Hamiltonian path has a TSP tour of length at most 4n/3. Since a Hamiltonian path is a spanning tree with two leaves, this motivates the question of whether or not a graph containing a spanning tree with few leaves has a short TSP tour. The recent techniques of Mömke and Svensson imply that a graph containing a depth-first-search tree with k leaves has a TSP tour of length 4n/3 + O(k). Using our approach, we can show that a 2(k − 1)-vertex connected graph that contains a spanning tree with at most k leaves has a TSP tour of length 4n/3. We also explore other conditions under which our approach results in a short tour.

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