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A note on the Song-Zhang theorem for Hamiltonian graphs

Kewen ZhaoRonald J. Gould — 2010

Colloquium Mathematicae

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. In 1994, Song and Zhang proved that if for each independent set S of cardinality k+1, one of the following condition holds: (i) there exist u ≠ v ∈ S such that d(u) + d(v) ≥ n or |N(u) ∩ N(v)| ≥ α (G); (ii) for any distinct u and v in S, |N(u) ∪ N(v)| ≥ n - max{d(x): x ∈ S}, then G is Hamiltonian. We prove that if for each essential...

Weak Saturation Numbers for Sparse Graphs

Ralph J. FaudreeRonald J. GouldMichael S. Jacobson — 2013

Discussiones Mathematicae Graph Theory

For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order...

On the Non-(p−1)-Partite Kp-Free Graphs

Kinnari AminJill FaudreeRonald J. GouldElżbieta Sidorowicz — 2013

Discussiones Mathematicae Graph Theory

We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?

2-factors in claw-free graphs

Guantao ChenJill R. FaudreeRonald J. GouldAkira Saito — 2000

Discussiones Mathematicae Graph Theory

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 K 1 , 3 .) 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.

Forbidden triples implying Hamiltonicity: for all graphs

Ralph J. FaudreeRonald J. GouldMichael S. Jacobson — 2004

Discussiones Mathematicae Graph Theory

In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with G i = K 1 , 3 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 K 1 , s , 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 K 1 , 3 , such that all G₁G₂G₃-free...

Potential forbidden triples implying hamiltonicity: for sufficiently large graphs

Ralph J. FaudreeRonald J. GouldMichael S. Jacobson — 2005

Discussiones Mathematicae Graph Theory

In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with G i = K 1 , 3 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 K 1 , s , 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 K 1 , 3 , such that all G₁G₂G₃-free graphs are...

The structure and existence of 2-factors in iterated line graphs

Michael FerraraRonald J. GouldStephen G. Hartke — 2007

Discussiones Mathematicae Graph Theory

We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where L k ( G ) contains a 2-factor.

The Chvátal-Erdős condition and 2-factors with a specified number of components

Guantao ChenRonald J. GouldKen-ichi KawarabayashiKatsuhiro OtaAkira SaitoIngo Schiermeyer — 2007

Discussiones Mathematicae Graph Theory

Let G be a 2-connected graph of order n satisfying α(G) = a ≤ κ(G), where α(G) and κ(G) are the independence number and the connectivity of G, respectively, and let r(m,n) denote the Ramsey number. The well-known Chvátal-Erdös Theorem states that G has a hamiltonian cycle. In this paper, we extend this theorem, and prove that G has a 2-factor with a specified number of components if n is sufficiently large. More precisely, we prove that (1) if n ≥ k·r(a+4, a+1), then G has a 2-factor with k components,...

Chvátal-Erdös type theorems

Jill R. FaudreeRalph J. FaudreeRonald J. GouldMichael S. JacobsonColton Magnant — 2010

Discussiones Mathematicae Graph Theory

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-k)/(k+1),...

Saturation Spectrum of Paths and Stars

Jill FaudreeRalph J. FaudreeRonald J. GouldMichael S. JacobsonBrent J. Thomas — 2017

Discussiones Mathematicae Graph Theory

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from G̅ to G results in a copy of H. The minimum size of an H-saturated graph on n vertices is denoted sat(n,H), while the maximum size is the well studied extremal number, ex(n,H). The saturation spectrum for a graph H is the set of sizes of H saturated graphs between sat(n,H) and ex(n,H). In this paper we completely determine the saturation spectrum of stars and we show the saturation spectrum of paths is continuous...

Linear forests and ordered cycles

Guantao ChenRalph J. FaudreeRonald J. GouldMichael S. JacobsonLinda LesniakFlorian Pfender — 2004

Discussiones Mathematicae Graph Theory

A collection L = P ¹ P ² . . . P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian.

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