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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A note on a new condition implying pancyclism

100%
Flandrin E., Li H., Marczyk A., Woźniak M.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 1
137-143
EN
We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n/2,n/2}$.
2
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Pancyclism and small cycles in graphs

100%
Faudree R., Favaron O., Flandrin E., Li H.
Discussiones Mathematicae Graph Theory
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1996
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tom 16
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nr 1
27-40
EN
We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u)+d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u)+d(v) ≥ n+z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (d_C(u,v)+1, [(n+19)/13]), $d_C(u,v)$ being the distance of u and v on a hamiltonian cycle of G.
3
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An Implicit Weighted Degree Condition For Heavy Cycles

100%
Cai J., Li H., Ning W.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 4
801-810
EN
For a vertex v in a weighted graph G, idw(v) denotes the implicit weighted degree of v. In this paper, we obtain the following result: Let G be a 2-connected weighted graph which satisfies the following conditions: (a) The implicit weighted degree sum of any three independent vertices is at least t; (b) w(xz) = w(yz) for every vertex z ∈ N(x) ∩ N(y) with xy /∈ E(G); (c) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a hamiltonian cycle or a cycle of weight at least 2t/3. This generalizes the result of Zhang et al. [9].
4
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A Note on Neighbor Expanded Sum Distinguishing Index

88%
Flandrin E., Li H., Marczyk A., Saclé J.-F., Woźniak M.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
29-37
EN
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.
5
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Chvátal-Erdos condition and pancyclism

88%
Flandrin E., Li H., Marczyk A., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
335-342
EN
The well-known Chvátal-Erdős theorem states that if the stability number α of a graph G is not greater than its connectivity then G is hamiltonian. In 1974 Erdős showed that if, additionally, the order of the graph is sufficiently large with respect to α, then G is pancyclic. His proof is based on the properties of cycle-complete graph Ramsey numbers. In this paper we show that a similar result can be easily proved by applying only classical Ramsey numbers.
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