3-connected Claw-free Graph Research Articles

Hamiltonicity and Minimum Degree in 3-connected Claw-Free Graphs

Using Ryjáček's closure, we prove that any 3-connected claw-free graph of order ν and minimum degree δ⩾ν+3810 is hamiltonian. This improves a theorem of Kuipers and Veldman who got the same result with the stronger hypotheses δ⩾ν+298 and ν sufficiently large and nearly proves their conjecture saying that the result holds as soon as δ⩾ν+610 for ν sufficiently large.

Clique covering and degree conditions for hamiltonicity in claw-free graphs

By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian claw-free graph G satisfying a degree condition of type σ k ( G)> n+ k 2−4 k+7 (where k is a constant), the closure of G can be covered by at most k−1 cliques. Using structural properties of the closure concept, we show a method for characterizing all such nonhamiltonian exceptional graphs with limited clique covering number. The method is explicitly carried out for k⩽6 and illustrated by proving that every 2-connected claw-free graph G of order n⩾77 with δ( G)⩾14 and σ 6( G)> n+19 is either hamiltonian or belongs to a family of easily described exceptions.

Nonhamiltonian 2-connected claw-free graphs with large 4-degree sum

Let G be a 2-connected claw-free graph on n vertices. Let σ k ( G) be the minimum degree sum among k-element independent set of vertices in G. It is proved that if σ 4( G)⩾ n+3 then G is hamiltonian or else G belong to the known family of graphs. This is a generalization of the best known sufficient condition on hamiltonicity in claw-free 2-connected graphs given independently by Liu, Zhang and Broersma. Moreover, it is shown that the problem HAMILTONIAN CYCLE restricted to claw-free graphs G=( V, E) with σ 3(G)⩾⌊ 3 4 (|G|+3)⌋ has polynomial time complexity. This contrasts sharply with known results on NP-completeness among dense graphs.

A nine vertex theorem for 3-connected claw- free graphs

It is proved that if G is a 3-connected claw-free graph and if S is a set of vertices in G with j S j 5 9, then G contains a cycle which in turn contains S. This result is sharp.

A Fan-type condition for claw-free graphs to be Hamiltonian

Let G be a 2-connected claw-free graph of order n. If for each pair of vertices u and v dist(u,v)=3⇒ max(d(u),d(v))⩾(n−4)/2, then G is Hamiltonian.

Linear Time Algorithms for Hamiltonian Problems on (Claw,Net)-Free Graphs

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)-free graphs. These graphs enjoy those interesting structural properties.

Minimal 2-connected non-hamiltonian claw-free graphs

We say a graph G is minimal with respect to a property Q if there exists no proper induced subgraph G′ of G with property Q. In this paper we characterize all minimal graphs with respect to the property ‘to be 2-connected, non-hamiltonian and claw-free’. Several sufficient forbidden subgraph conditions are obtained as corollaries.

Vertex pancyclism in claw-free graphs

In this paper, we show four sufficient conditions for vertex pancyclicity in claw-free graphs. These results improve, respectively, results of Broersma and Veldman, Gould and Jacobson, and Ryjacek. In particular, we prove that if G is a 3-connected claw-free graph such that the vertices of degree 1 of every induced bull have a common neighbor in G then every vertex of G is contained in a cycle of all possible lengths except for four and five. We conjecture that almost all pancyclic claw-free graphs with minimum degree at least three are vertex pancyclic.

Hamiltonicity in 2-connected graphs with claws

Matthews and Sumner have proved in [12] that if G is a 2-connected claw-free graph of order n such that δ ⩾ ( n − 2)/3, then G is hamiltonian. Li has shown that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in Π, where Π is a class of graphs defined in [7]. On the other hand, we say that a graph G is almost claw-free if the centres of induced claws are independent and their neighbourhoods are 2-dominated. Broersma, Ryjáček and Schiermeyer have proved that if G is 2-connected almost claw-free graph of order n such that δ ⩾ ( n − 2)/3, then G is hamiltonian. We generalize these results by considering the graphs whose claw centres are independent. If G is a 2-connected graph of order n and minimum degree δ such that n ⩽ 4 δ − 3 and if the set of claw centres of G is independent, then we show that either G is hamiltonian or G ϵ F, where F is a class of graphs defined in the paper. The bound n ⩽ 4 δ − 3 is sharp.

Claw-free graphs — A survey

In this paper we summarize known results on claw-free graphs. The paper is subdivided into the following chapters and sections: 1. 1. Introduction 2. 2. Paths, cycles, hamiltonicity 2.1. (a) Preliminaries 2.2. (b) Degree and neighborhood conditions 2.3. (c) Local connectivity conditions 2.4. (d) Further forbidden subgraphs 2.5. (e) Invariants 2.6. (f) Squares 2.7. (g) Regular graphs 2.8. (h) Other hamiltonicity related results and generalizations 2.9. 3. Matchings and factors 2.10. 4. Independence, domination, other invariants and extremal problems 2.11. 5. Algorithmic aspects 2.12. 6. Miscellaneous 2.13. 7. Appendix — List of all 2-connected nonhamiltonian claw-free graphs on n ⩽ 12 vertices.

Hamiltonian cycles in regular 3-connected claw-free graphs

A known result by the author in 1991 is that every 3-connected claw-free graph on at most 6δ-11 vertices is Hamiltonian. In this paper it is proved that every 3-connected k-regular claw-free graph on at most 7 k — 19 vertices is Hamiltonian.

Hamiltonicity in claw-free graphs through induced bulls

In the present paper we show that if G is a 2-connected claw-free graph such that the vertices of degree 1 of every induced bull have a common neighbour in G then G is hamiltonian. This statement was originally conjectured by H.J. Broersma and H.J. Veldman

Longest cycles in regular 2-connected claw-free graphs

In this paper, we show that every 2-connected, k-regular claw-free graph on n vertices contains a cycle of length at least min 4k−2, n (k⩾8), and this result is best possible.

Extending matchings in claw-free graphs

Let G be a graph with a perfect matching and let n be an integer, 1⩽ n<| V( G)|⧸2. Graph G is n- extendable if every matching of size n in G is a subset of a perfect matching. Graph G is bicritical if G— u— v has a perfect matching for every pair of points u and v in V( G). It is proved that every 3-connected claw-free graph is bicritical and for n⩾2, every (2 n+1)-connected claw-free graph is n-extendable. Matching extension in planar and toroidal claw-free graphs is then considered.

On hamiltonian claw-free graphs

We show that every 3-connected claw-free graphs having at most 5δ – 10 vertices is hamiltonian, where δ is the minimum degree. For regular 3-connected claw-free graphs, a related result was obtained by Li and Liu (preprint), but for nonregular claw-free graphs the best-known result comes from the work of Zhang (1988), with n⩽4δ+3.

Hamiltonicity in claw-free graphs

A graph is claw-free if it contains no induced subgraph isomorphic to a K 1,3. This paper studies hamiltonicity in two subclasses of claw-free graphs. A claw-free graph is CN-free (claw-free, net-free) if it does not contain an induced subgraph isomorphic to a net (a triangle with a pendant leaf dangling from each vertex). We give a structural characterisation of CN-free graphs which yields a simple proof that any connected (1-tough) CN-free graph has a Hamilton path (cycle). We strengthen this result for CN-free graphs of higher connectivity. For example, we show 3-connected CN-free graphs are both Hamilton connected and pancyclic. In addition, the characterisation yields a polynomial algorithm for finding a Hamilton path (cycle) in a connected (2-connected) CN-free graph. The second class consists of those claw-free graphs such that for each vertex v the set of vertices at distance two from v does not contain an independent subset of size three. We call such graphs distance claw-free and give a forbidden subgraph characterisation. We use this along with our earlier characterisation to prove that any 2-connected ( 3 2 - tough ) distance claw-free graph has a Hamilton path (cycle). Our characterisation also gives a polynomial time algorithm to solve the weighted stable set problem in a distance claw-free graph.