Hamiltonicity Of Claw-free Graphs Research Articles

Induced Nets and Hamiltonicity of Claw-Free Graphs

The connected graph of degree sequence 3, 3, 3, 1, 1, 1 is called a net, and the vertices of degree 1 in a net are called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced \(K_{1,3}\) is hamiltonian if every endvertex of each induced net of G has degree at least \((|V(G)|-2)/3\). In this paper we prove this conjecture in the affirmative.

A Note on Singular Edges and Hamiltonicity in Claw-Free Graphs with Locally Disconnected Vertices

An edge e of a graph G is called singular if it is not on a triangle; otherwise, e is nonsingular. A vertex is called singular if it is adjacent to a singular edge; otherwise, it is called nonsingular. We prove the following. Let G be a connected claw-free graph such that every locally disconnected vertex x ∈ V(G) satisfies the following conditions: Then G is either hamiltonian, or G is the line graph of the graph obtained from $$K_{2,3}$$ by attaching a pendant edge to its each vertex of degree two. Some results on forbidden subgraph conditions for hamiltonicity in 3-connected claw-free graphs are also obtained as immediate corollaries

Hamiltonicity of claw-free graphs and Fan-type conditions

For a graph H, let δF(H)=min{max{d(u),d(v)}|for any u,v∈V(H) with distH(u,v)=2}. For a given integer p>0 and a given number ϵ, a graph H of order n is said to satisfy a Fan-type condition if δF(H)≥2n+ϵp. In this paper, we show that like the planar graphs having obstruction set {K5,K3,3}, for the family of k-connected claw-free Hamiltonian graphs H with δF(H)≥2n+ϵp, if k=3 it has a finite obstruction set in which each graph has order at most max{9p∕2−20,10}, and if k=2 it has an obstruction set in which each graph has at most max{9p−20,0} vertices of degree greater than 3. We determine the best possible values of p and ϵ when the obstruction set is empty. We show that for k-connected claw-free simple graphs H of order n with n large enough, each of the following holds:(a) if k=2 and δF(H)≥2n−114 then H is Hamiltonian;(b) if k=3 and δF(H)≥2n−198 then H is Hamiltonian.These bounds are sharp. For k=3 and a given integer p, since the obstruction set is finite and can be determined in polynomial time, improvements to (b) for other given values of p are possible with the help of a computer.

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erdős proved that if a graph G with n vertices satisfies e(G)>maxn−k2+k2,⌈(n+1)∕2⌉2+n−122,where the minimum degree δ(G)≥k and 1≤k≤(n−1)∕2, then it is Hamiltonian. For n≥2k+1, let Enk=Kk∨(kK1+Kn−2k), where “∨” is the “join” operation. One can observe e(Enk)=n−k2+k2 and Enk is not Hamiltonian. As Enk contains induced claws for k≥2, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with the largest possible number of edges. We answer this question completely by proving a claw-free analog of Erdős’ theorem. Moreover, as byproducts, we establish several tight spectral conditions for a 2-connected claw-free graph to be Hamiltonian. Similar results for the traceability of connected claw-free graphs are also obtained. Our tools include Ryjáček’s claw-free closure theory and Brousek’s characterization of minimal 2-connected claw-free non-Hamiltonian graphs.

Neighborhood union of independent sets and hamiltonicity of claw-free graphs

Let G be a graph, for any u∈V(G), let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U⊆V(G), let N(U)=∪ u eU N(u), and d(U)=|N(U)|. A graph G is called claw-free if it has no induced subgraph isomorphic to K 1,3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng, et al.: Let G be a 2-connected claw-free graph of order n, and d(u)+d(v)+d(w)≥n−2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian.

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.

On a Closure Concept in Claw-Free Graphs

IfGis a claw-free graph, then there is a graphcl(G) such that (i)Gis a spanning subgraph ofcl(G), (ii)cl(G) is a line graph of a triangle-free graph, and (iii)the length of a longest cycle inGand incl(G) is the same. A sufficient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected claw-free graphs are obtained as corollaries.

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

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.