On the number of components in 2-factors of claw-free graphs

In this paper, we prove that if a claw-free graph G with minimum degree δ?4 has no maximal clique of two vertices, then G has a 2-factor with at most (|G|-1)/4 components. This upper bound is best possible. Additionally, we give a family of claw-free graphs with minimum degree δ?4 in which every 2-factor contains more than n/δ components.     

Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs.     

Hamiltonicity of 6-connected line graphs

Let G be a graph and let D₆(G)={v∈V(G)|d_G(v)=6}. In this paper we prove that: (i) If G is a 6-connected claw-free graph and if |D₆(G)|≤74 or G[D₆(G)] contains at most 8 vertex disjoint K₄’s, then G is Hamiltonian; (ii) If G is a 6-connected line graph and if |D₆(G)|≤54 or G[D₆(G)] contains at most 5 vertex disjoint K₄’s, then G is Hamilton-connected.     

The asymptotic number of claw-free cubic graphs

Let H_n be the number of claw-free cubic graphs on 2n labeled nodes. In an earlier paper we characterized claw-free cubic graphs and derived a recurrence relation for H_n. Here we determine the asymptotic behavior of this sequence:

     

Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑_v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z₃ denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z₃-connected. In this paper, we proved the following.

• (i) 

  Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.

• (ii) 

  Every 6-edge-connected triangular line graph is Z3-connected.

• (iii) 

  Every 7-edge-connected triangular claw-free graph is Z3-connected.

In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.     

Path extendability of claw-free graphs

Let G be a connected, locally connected, claw-free graph of order n and x,y be two vertices of G. In this paper, we prove that if for any 2-cut S of G, S∩{x,y}=∅, then each (x,y)-path of length less than n-1 in G is extendable, that is, for any path P joining x and y of length h(<n-1), there exists a path P^′ in G joining x and y such that V(P)⊂V(P^′) and |P^′|=h+1. This generalizes several related results known before.     

Counting labeled claw-free cubic graphs by connectivity

Claw-free cubic graphs are counted with given connectedness and order. Tables are provided for claw-free cubic graphs with given connectedness. This builds on methods for counting general cubic graphs by connectivity previously developed by Chae, Palmer, and Robinson, and on the earlier enumeration of all claw-free cubic graphs by McKay, Palmer, Read, and Robinson.     

On the Zagreb indices of the line graphs of the subdivision graphs

The aim of this paper is to investigate the Zagreb indices of the line graphs of the tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts.     

On stable cutsets in claw-free graphs and planar graphs

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let $K_4$ and $K_{1,3}$ (claw) denote the complete (bipartite) graph on 4 and $1+3$ vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a $K_4$-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, $K_4$)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for $K_4$-free planar graphs with maximum degree five.     

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erd?s proved that if a graph $G$ with $n$ vertices satisfies

$e\left(G\right)>max\left\{\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2,\left(\genfrac{}{}{0ex}{}{?(n+1)∕2?}{2}\right)+{?\frac{n?1}{2}?}^2\right\},$

where the minimum degree $\delta \left(G\right)\ge k$ and $1\le k\le (n?1)∕2$, then it is Hamiltonian. For $n\ge 2k+1$, let $E_n^k=K_k\vee (k{K}_1+K_{n?2k})$, where “$\vee$” is the “join” operation. One can observe $e\left(E_n^k\right)=\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2$ and $E_n^k$ is not Hamiltonian. As $E_n^k$ contains induced claws for $k\ge 2$, 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 Erd?s’ 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 Ryjá?ek’s claw-free closure theory and Brousek’s characterization of minimal 2-connected claw-free non-Hamiltonian graphs.