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Longest cycles and their chords

✍ Scribed by Cun-Quan Zhang


Publisher
John Wiley and Sons
Year
1987
Tongue
English
Weight
386 KB
Volume
11
Category
Article
ISSN
0364-9024

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✦ Synopsis


Thomassen conjectured that any longest cycle of a 3-connected graph has a chord. In this paper, we will show that the conjecture is true for a planar graph if it is cubic or 6 2 4.


πŸ“œ SIMILAR VOLUMES


Chords of Longest Cycles in Cubic Graphs
✍ Carsten Thomassen πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 213 KB

We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and

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Using a variation of Thomassen's admissible triples technique, we give an alternative proof that every strongly 2-arc-connected directed graph with two or more vertices contains a directed cycle that has at least two chords, while at the same time establishing a more general result.

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## Abstract Let $n\_1,n\_2,\ldots,n\_k$ be integers, $n=\sum n\_i$, $n\_i\ge 3$, and let for each $1\le i\le k$, $H\_i$ be a cycle or a tree on $n\_i$ vertices. We prove that every graph __G__ of order at least __n__ with $\sigma\_2(G) \ge 2( n-k) -1$ contains __k__ vertex disjoint subgraphs $H\_1'

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In this article, we establish bounds for the length of a longest cycle C in a 2-connected graph G in terms of the minimum degree Ξ΄ and the toughness t. It is shown that C is a Hamiltonian cycle or |C| β‰₯ (t + 1)Ξ΄ + t.

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In this paper we obtain two sufficient conditions, Ore type (Theorem 1) and Dirac type (Theorem 2). on the degrees of a bipartite oriented graph for ensuring the existence of long paths and cycles. These conditions are shown to be the best possible in a sense. An oriented graph is a digraph without

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## Abstract For a graph __G__, __p__(__G__) and __c__(__G__) denote the order of a longest path and a longest cycle of __G__, respectively. In this paper, we prove that if __G__ is a 3 ‐connected graph of order __n__ such that the minimum degree sum of four independent vertices is at least __n__+ 6