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Tytuł:
Extension of several sufficient conditions for Hamiltonian graphs
Autorzy:
Ainouche, Ahmed
Powiązania:
https://bibliotekanauki.pl/articles/744192.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hamiltonian graph
dual closure
neighborhood closure
Opis:
Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖{u})|+d(u) ≥ n-1. Using the concept of dual closure, we prove that
1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇
2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_r ∪ Kₛ ∪ Kₜ) ∨ K₂$, 1 ≤ r ≤ s ≤ t or the graph $((n+1)/2)K₁ ∨ K_{(n-1)/2}$.
It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 1; 23-39
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Variations on a sufficient condition for Hamiltonian graphs
Autorzy:
Ainouche, Ahmed
Lapiquonne, Serge
Powiązania:
https://bibliotekanauki.pl/articles/743758.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
cycles
partially square graph
degree sum
independent sets
neighborhood unions and intersections
dual closure
Opis:
Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ {x}$. In particular, this condition is satisfied if x does not center a claw (an induced $K_{1,3}$). Clearly G ⊆ G* ⊆ G², where G² is the square of G. For any independent triple X = {x,y,z} we define
σ̅(X) = d(x) + d(y) + d(z) - |N(x) ∩ N(y) ∩ N(z)|.
Flandrin et al. proved that a 2-connected graph G is hamiltonian if [σ̅]₃(X) ≥ n holds for any independent triple X in G. Replacing X in G by X in the larger graph G*, Wu et al. improved recently this result. In this paper we characterize the nonhamiltonian 2-connected graphs G satisfying the condition [σ̅]₃(X) ≥ n-1 where X is independent in G*. Using the concept of dual closure we (i) give a short proof of the above results and (ii) we show that each graph G satisfying this condition is hamiltonian if and only if its dual closure does not belong to two well defined exceptional classes of graphs. This implies that it takes a polynomial time to check the nonhamiltonicity or the hamiltonicity of such G.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 2; 229-240
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Nested Locally Hamiltonian Graphs and the Oberly-Sumner Conjecture
Autorzy:
de Wet, Johan P.
Frick, Marietjie
Powiązania:
https://bibliotekanauki.pl/articles/32222536.pdf
Data publikacji:
2022-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
locally traceable
locally hamiltonian
Hamilton Cycle Problem
locally k -nested-hamiltonian
Oberly-Sumner Conjecture
Opis:
A graph G is locally P, abbreviated L, if for every vertex v in G the open neighbourhood N(v) of v is non-empty and induces a graph with property P. Specifically, a graph G without isolated vertices is locally connected (LC) if N(v) induces a connected graph for each v ∈ V (G), and locally hamiltonian (LH) if N(v) induces a hamiltonian graph for each v ∈ V (G). A graph G is locally locally P (abbreviated L2P) if N(v) is non-empty and induces a locally P graph for every v ∈ V (G). This concept is generalized to an arbitrary degree of nesting. For any k ≥ 0 we call a graph locally k-nested-hamiltonian if it is LmC for m = 0, 1, . . ., k and LkH (with L0C and L0H meaning connected and hamiltonian, respectively). The class of locally k-nested-hamiltonian graphs contains important subclasses. For example, Skupień had already observed in 1963 that the class of connected LH graphs (which is the class of locally 1-nested-hamiltonian graphs) contains all triangulations of closed surfaces. We show that for any k ≥ 1 the class of locally k-nested-hamiltonian graphs contains all simple-clique (k + 2)-trees. In 1979 Oberly and Sumner proved that every connected K1,3-free graph that is locally connected is hamiltonian. They conjectured that for k ≥ 1, every connected K1,k+3-free graph that is locally (k + 1)-connected is hamiltonian. We show that locally k-nested-hamiltonian graphs are locally (k + 1)-connected and consider the weaker conjecture that every K1,k+3-free graph that is locally k-nested-hamiltonian is hamiltonian. We show that if our conjecture is true, it would be “best possible” in the sense that for every k ≥ 1 there exist K1,k+4-free locally k-nested-hamiltonian graphs that are non-hamiltonian. We also attempt to determine the minimum order of non-hamiltonian locally k-nested-hamiltonian graphs and investigate the complexity of the Hamilton Cycle Problem for locally k-nested-hamiltonian graphs with restricted maximum degree.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 4; 1281-1312
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Hamiltonian Extendable Graphs
Autorzy:
Yang, Xiaojing
Xiong, Liming
Powiązania:
https://bibliotekanauki.pl/articles/32304150.pdf
Data publikacji:
2022-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Hamiltonian extendable
forbidden subgraph
Opis:
A graph is called Hamiltonian extendable if there exists a Hamiltonian path between any two nonadjacent vertices. In this paper, we give an explicit formula of the minimum number of edges for Hamiltonian extendable graphs and we also characterize the degree sequence for Hamiltonian extendable graphs with minimum number of edges. Besides, we completely characterize the pairs of forbidden subgraphs for 2-connected graphs to be Hamiltonian extendable.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 3; 843-859
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the H -Force Number of Hamiltonian Graphs and Cycle Extendability
Autorzy:
Hexel, Erhard
Powiązania:
https://bibliotekanauki.pl/articles/31342172.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
cycle
hamiltonian graph
H -force number
cycle extendability
Opis:
The H-force number h(G) of a hamiltonian graph G is the smallest cardinality of a set A ⊆ V (G) such that each cycle containing all vertices of A is hamiltonian. In this paper a lower and an upper bound of h(G) is given. Such graphs, for which h(G) assumes the lower bound are characterized by a cycle extendability property. The H-force number of hamiltonian graphs which are exactly 2-connected can be calculated by a decomposition formula.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 79-88
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
New sufficient conditions for hamiltonian and pancyclic graphs
Autorzy:
Schiermeyer, Ingo
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/743639.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hamiltonian graphs
pancyclic graphs
closure
Opis:
For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 1; 29-38
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on a new condition implying pancyclism
Autorzy:
Flandrin, Evelyne
Li, Hao
Marczyk, Antoni
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/743445.pdf
Data publikacji:
2001
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hamiltonian graphs
pancyclic graphs
cycles
Opis:
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 $K_{n/2,n/2}$.
Źródło:
Discussiones Mathematicae Graph Theory; 2001, 21, 1; 137-143
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Heavy Subgraphs, Stability and Hamiltonicity
Autorzy:
Li, Binlong
Ning, Bo
Powiązania:
https://bibliotekanauki.pl/articles/31341693.pdf
Data publikacji:
2017-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
heavy subgraphs
hamiltonian graphs
closure theory
Opis:
Let G be a graph. Adopting the terminology of Broersma et al. and Čada, respectively, we say that G is 2-heavy if every induced claw (K1,3) of G contains two end-vertices each one has degree at least |V (G)|/2; and G is o-heavy if every induced claw of G contains two end-vertices with degree sum at least |V (G)| in G. In this paper, we introduce a new concept, and say that G is S-c-heavy if for a given graph S and every induced subgraph G′ of G isomorphic to S and every maximal clique C of G′, every non-trivial component of G′ − C contains a vertex of degree at least |V (G)|/2 in G. Our original motivation is a theorem of Hu from 1999 that can be stated, in terms of this concept, as every 2-connected 2-heavy and N-c-heavy graph is hamiltonian, where N is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs S such that every 2-connected o-heavy and S-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu’s theorem. Furthermore, our main result improves or extends several previous results.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 3; 691-710
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The flower conjecture in special classes of graphs
Autorzy:
Ryjáček, Zdeněk
Schiermeyer, Ingo
Powiązania:
https://bibliotekanauki.pl/articles/972046.pdf
Data publikacji:
1995
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hamiltonian graphs
flower conjecture
square
claw-free graphs
Opis:
We say that a spanning eulerian subgraph F ⊂ G is a flower in a graph G if there is a vertex u ∈ V(G) (called the center of F) such that all vertices of G except u are of the degree exactly 2 in F. A graph G has the flower property if every vertex of G is a center of a flower. Kaneko conjectured that G has the flower property if and only if G is hamiltonian. In the present paper we prove this conjecture in several special classes of graphs, among others in squares and in a certain subclass of claw-free graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 1995, 15, 2; 179-184
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Hamiltonian Normal Cayley Graphs
Autorzy:
Montellano-Ballesteros, Juan José
Arguello, Anahy Santiago
Powiązania:
https://bibliotekanauki.pl/articles/31343293.pdf
Data publikacji:
2019-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Cayley graph
hamiltonian cycle
normal connection set
Opis:
A variant of the Lovász Conjecture on hamiltonian paths states that every finite connected Cayley graph contains a hamiltonian cycle. Given a finite group G and a connection set S, the Cayley graph Cay(G, S) will be called normal if for every g ∈ G we have that g−1Sg = S. In this paper we present some conditions on the connection set of a normal Cayley graph which imply the existence of a hamiltonian cycle in the graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 3; 731-740
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Chvátal-Erdos condition and pancyclism
Autorzy:
Flandrin, Evelyne
Li, Hao
Marczyk, Antoni
Schiermeyer, Ingo
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/743987.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
hamiltonian graphs
pancyclic graphs
cycles
connectivity
stability number
Opis:
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.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 2; 335-342
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs
Autorzy:
Seamone, Ben
Powiązania:
https://bibliotekanauki.pl/articles/31339495.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Hamiltonian cycle
uniquely Hamiltonian graphs
claw-free graphs
triangle-free graphs
Opis:
A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 207-214
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
2-Connected Hamiltonian Claw-Free Graphs Involving Degree Sum of Adjacent Vertices
Autorzy:
Tian, Tao
Xiong, Liming
Powiązania:
https://bibliotekanauki.pl/articles/32034090.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Hamiltonian cycle
degree sum
dominating closed trail
closure
Opis:
For a graph H, define $\overline{\sigma}_2(H)=min\{d(u)+d(v)|uv∈E(H)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with \(\delta(H) ≥ 3\). In [J. Graph Theory 86 (2017) 193–212], Chen proved that if $\overline{\sigma}_2(H)≥\frac{n}{2}−1$ and $n$ is sufficiently large, then $H$ is Hamiltonian with two families of exceptions. In this paper, we refine the result. We focus on the condition $\overline{\sigma}_2(H)≥\frac{2n}{5}−1$, and characterize non-Hamiltonian 2-connected claw-free graphs $H$ of order $n$ sufficiently large with $\overline{\sigma}_2(H)≥\frac{2n}{5}−1$. As byproducts, we prove that there are exactly six graphs in the family of 2-edge-connected triangle-free graphs of order at most seven that have no spanning closed trail and give an improvement of a result of Veldman in [Discrete Math. 124 (1994) 229–239].
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 85-106
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Hamiltonian and Pancyclic Graphs in the Class of Self-Centered Graphs with Radius Two
Autorzy:
Hrnčiar, Pavel
Monoszová, Gabriela
Powiązania:
https://bibliotekanauki.pl/articles/31342283.pdf
Data publikacji:
2018-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
self-centered graph with radius 2
Hamiltonian graph
pancyclic graph
size of graph
Opis:
The paper deals with Hamiltonian and pancyclic graphs in the class of all self-centered graphs of radius 2. For both of the two considered classes of graphs we have done the following. For a given number n of vertices, we have found an upper bound of the minimum size of such graphs. For n ≤ 12 we have found the exact values of the minimum size. On the other hand, the exact value of the maximum size has been found for every n. Moreover, we have shown that such a graph (of order n and) of size m exists for every m between the minimum and the maximum size. For n ≤ 10 we have found all nonisomorphic graphs of the minimum size, and for n = 11 only for Hamiltonian graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 3; 661-681
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on Hamiltonian cycles in generalized Halin graphs
Autorzy:
Bojarska, Magdalena
Powiązania:
https://bibliotekanauki.pl/articles/744110.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graphs
Halin graphs
hamiltonian cycles
Opis:
We show that every 2-connected (2)-Halin graph is Hamiltonian.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 4; 701-704
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 35

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