Journal ArticleDOI
Long paths and large cycles in finite graphs
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Ore-type sufficient conditions ensuring the existence of a large cycle passing through any given path of length s for (s + 2)-connected graphs are given, and the extremal cases are characterized.Abstract:
Ore-type sufficient conditions ensuring the existence of a large cycle passing through any given path of length s for (s + 2)-connected graphs are given, and the extremal cases are characterized.read more
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Journal ArticleDOI
Heavy paths and cycles in weighted graphs
TL;DR: It is proved that for a 2-connected weighted graph, if every vertex has weighted degree at least d, then for any given vertex y, either y is contained in a cycle with weight at least 2d or every heaviest cycle is a Hamilton cycle.
Journal Article
Spanning trees with a bounded number of leaves in a claw-free graph
TL;DR: It is proved that if a connected claw-free graph G satisfies σk+1(G) ≥ |G| − k, then G has a spanning tree with at most k leaves and the bound |G | − k is sharp.
Journal ArticleDOI
A generalization of dirac’s theorem
TL;DR: LetG be an (r+2)-connected graph in which every vertex has degree at leastd and which has at least 2d-r vertices and for any pathQ of lengthr and vertexy not onQ, there is a cycle of length at least2d- r containing bothQ andy.
Journal ArticleDOI
Stability in the Erdős–Gallai Theorem on cycles and paths, II
TL;DR: A stability theorem is complete which strengthens Kopylov’s result that for k ≥ 3 odd and all n ≥ k, every n -vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs.
Journal ArticleDOI
Stability in the Erdős–Gallai Theorems on cycles and paths
TL;DR: In this article, it was shown that for all n ≥ 3 t > 3, and k ∈ { 2 t + 1, 2 t+2 }, every n -vertex 2-connected graph G with e (G ) > h ( n, k, t − 1 ) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest.
References
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Book ChapterDOI
On Maximal Circuits in Finite Graphs
TL;DR: In this paper, the authors focus on the following refinement: if d(v + d(w) ≥ n(G) > 2 for any two different, nonadjacent vertices v, w of G, then G contains a hamiltonian circuit.
Book ChapterDOI
Graphs with Cycles Containing Given Paths
TL;DR: In this article, it was shown that given any path of length r there is a cycle of length at least m ≥ r + 3 containing this path, which implies the well-known theorem of Chvatal [4] on hamiltonian graphs and the theorem of Posa [7] which gives sufficient conditions for a graph to contain cycles of a certain length.