Shinya Fujita (2005)
Discussiones Mathematicae Graph Theory
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Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.
Guantao Chen, Jill R. Faudree, Ronald J. Gould, Akira Saito (2000)
Discussiones Mathematicae Graph Theory
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We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced .) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ (n-2)/3, G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ (n-24)/3. We also show that this result is sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k. ...
Raffaele Mosca (2012)
Discussiones Mathematicae Graph Theory
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The Maximum Stable Set (MS) problem is a well known NP-hard problem. However different graph classes for which MS can be efficiently solved have been detected and the augmenting graph technique seems to be a fruitful tool to this aim. In this paper we apply a recent characterization of minimal augmenting graphs [22] to prove that MS can be solved for -free graphs in polynomial time, extending some known results.
Zdeněk Ryjáček (1989)
Banach Center Publications
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Ahmed Ainouche, Serge Lapiquonne (2007)
Discussiones Mathematicae Graph Theory
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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 , where . In particular, this condition is satisfied if x does not center a claw (an induced ). 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...
Vadim E. Levit, Eugen Mandrescu (1998)
Discussiones Mathematicae Graph Theory
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The composition graph of a family of n+1 disjoint graphs is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978),...
Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2005)
Discussiones Mathematicae Graph Theory
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In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being , such that all G₁G₂G₃-free...
Hamamache Kheddouci (2001)
Discussiones Mathematicae Graph Theory
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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition , where . In the case where G is a claw-free graph, G* is equal to G². We define . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.
Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2004)
Discussiones Mathematicae Graph Theory
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In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁, G₂, G₃, none of which is a , s ≥ 3 such that G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G₁, G₂, G₃ with none being , such that all...
Ivan Havel, Bohdan Zelinka (2001)
Discussiones Mathematicae Graph Theory
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We deal with the graph operator defined to be the complement of the square of a graph: . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...
Bert Randerath, Preben Dahl Vestergaard (2002)
Discussiones Mathematicae Graph Theory
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A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...
Bert Randerath, Ingo Schiermeyer (2001)
Discussiones Mathematicae Graph Theory
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In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P₅-free or triangle-free, P₆-free and C₆-free graphs are 3-colourable. A canonical extension of these graph classes is , the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind,...
Ralph Faudree, András Gyárfás (1999)
Discussiones Mathematicae Graph Theory
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Let C denote the claw , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free...
Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)
Discussiones Mathematicae Graph Theory
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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...
Khadra Bouanane, Abdelhafid Berrachedi (2017)
Czechoslovak Mathematical Journal
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A -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free -graph. -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in -graphs and more specifically...
Ahmed Ainouche (2006)
Discussiones Mathematicae Graph Theory
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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 , 1 ≤ r ≤ s ≤ t or the graph . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity...
Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)
Discussiones Mathematicae Graph Theory
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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph induced by has property ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property...
Douglas R. Woodall (2011)
Discussiones Mathematicae Graph Theory
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It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ε, and k and d are positive integers such that k ≥ 3 and d is sufficiently large in terms of k and ε, then G is (k,d)*-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for...