3-connected Claw-free Graph Research Articles

On 2-factors with long cycles in 3-connected claw-free graphs

On 2-factors with long cycles in 3-connected claw-free graphs

Hamilton-connected claw-free graphs with Ore-degree conditions

Hamilton-connected claw-free graphs with Ore-degree conditions

Spanning trees with at most [formula omitted] leaves in 2-connected [formula omitted]-free graphs

Spanning trees with at most [formula omitted] leaves in 2-connected [formula omitted]-free graphs

The Independence Number Conditions for 2-Factors of a Claw-Free Graph

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices of degree not two, then we call it a branch. A set S of branches of a graph G is called a branch cut if we delete all edges and internal vertices of branches of S leading to more components than G. We use a branch bond to denote a minimal branch cut. If a branch-bond has an odd number of branches, then it is called odd. In this paper, we shall characterize all 2-connected claw-free graphs G such that every odd branch-bond of G has an edge branch and such that α(G)≤5 but has no 2-factor. We also consider the same problem for those 2-edge-connected claw-free graphs with α(G)≤4.

Every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected

We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.

On cycle-nice claw-free graphs

An even cycle C in a graph G is a nice cycle if G − V ( C ) has a perfect matching. A graph G is cycle-nice if each even cycle in G is a nice cycle. An even cycle C in an orientation of a graph G is clockwise odd if the number of its edges directed in the clockwise sense is odd. A graph G is Pfaffian if there is an orientation of G such that each nice cycle of G is clockwise odd. The significance of Pfaffian graphs is that the number of perfect matchings of a Pfaffian graph may be computed in polynomial time. Clearly, if G is a cycle-nice graph, then G is Pfaffian if and only if G admits an orientation such that each even cycle in G is clockwise odd. In this paper we obtain complete characterizations of 3-connected and 2-connected claw-free graphs that are cycle-nice. Using these characterizations, we can decide if a cycle-nice 2-connected claw-free graph is Pfaffian.

Hamiltonicity and restricted degree conditions on induced subgraphs in claw-free graphs, II

A wounded graph W is a graph obtained from a K 3 by adding a pendant edge at one vertex of the K 3 and adding a path of length 2 at another vertex of the K 3 . A net N is a graph obtained from a K 3 by adding a pendant edge at each vertex of the K 3 . A P 6 is a path on 6 vertices. Čada et al. (2016) conjectured that for a 2-connected claw-free graph H and for a fixed graph S ∈ { W , N , P 6 } , if the degree at each end-vertex of every induced copy of S is at least ( | V ( H ) | − 2 ) / 3 , then H is Hamiltonian. The case for S = N was conjectured by Broersma in [2] (1993). The conjecture for S ∈ { N , P 6 } was proved in [9] . In this paper, we prove the conjecture for the last case S = W . A new and shorter proof for the case S = N is also included.

Hamiltonicity and restricted degree conditions on induced subgraphs in claw-free graphs

For a graph S , a vertex with degree one in S is an end-vertex of S and an edge is a pendant edge if one of its ends is an end-vertex. A net N is a graph obtained from a K 3 by adding a pendant edge at each vertex of the K 3 . A P 6 is a path on 6 vertices. In this paper, we solve two conjectures and prove that for 2-connected claw-free graphs H and for a fixed graph S ∈ { N , P 6 } , if the degree at each end-vertex of every induced copy of S is at least ( | V ( H ) | − 2 ) ∕ 3 , then H is Hamiltonian. The case for S = N was conjectured by Broersma (1993); the case for S = P 6 was conjectured by C̃ada et al. (2016).

On sufficient degree conditions for traceability of claw-free graphs

We present new results on the traceability of claw-free graphs. In particular, we consider sufficient minimum degree and degree sum conditions that imply that these graphs admit a Hamilton path, unless they have a small order or they belong to well-defined classes of exceptional graphs. Our main result implies that a 2-connected claw-free graph G of sufficiently large order n with δ ( G ) ≥ 3 is traceable if the degree sum of any set of t independent vertices of G is at least t ( n + 6 ) 6 , where t ∈ { 1 , 2 , … , 6 } , and that this lower bound n + 6 6 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with minimum degree δ ( G ) ≥ 22 is traceable if the degree sum of any set of t independent vertices of G is at least t ( 2 n − 5 ) 14 , where t ∈ { 1 , 2 , … , 7 } , unless G is a member of well-defined classes of exceptional graphs depending on t , and that this lower bound 2 n − 5 14 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with δ ( G ) ≥ 18 is traceable if the degree sum of any set of 6 independent vertices is larger than n − 6 , and that this lower bound on the degree sums is sharp.

Cycle Traversability for Claw-Free Graphs and Polyhedral Maps

Let G be a graph, and $$v \in V(G)$$ and $$S \subseteq V(G)\setminus{v}$$ of size at least k. An important result on graph connectivity due to Perfect states that, if v and S are k-linked, then a (k−1)-link between a vertex v and S can be extended to a k-link between v and S such that the endvertices of the (k−1)-link are also the endvertices of the k-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets S1 and S2 are k-linked, then a t-link (t<k) between two disjoint sets S1 and S2 can be extended to a k-link between S1 and S2 such that the endvertices of the t-link are preserved in the k-link. Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices, but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that a 3-connected claw-free graph has a topological wheel minor Wk with k ≤ 5 if and only if it has a vertex of degree at least k. Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices, but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in surfaces having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.

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

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

The Hamilton-Connectivity with the Degree Sum of Non-adjacent Subgraphs of Claw-free Graphs

The degree d(H) of a subgraph H of a graph G is $$|\underset{u \in V(H)}{\cup} N(u)-V(H)|$$ , where N(u) denotes the neighbor set of the vertex u of G. In this paper, we prove the following result on the condition of the degrees of subgraphs. Let G be a 2-connected claw-free graph of order n with minimum degree δ(G) ≥ 3. If for any three non-adjacent subgraphs H1, H2, H3 that are isomorphic to K1, K1, K2, respectively, there is d(H1) + d(H2) + d(H3) ≥ n + 3, then for each pair of vertices u,v ∈ G that is not a cut set, there exists a Hamilton path between u and v.

On the independence number of traceable 2-connected claw-free graphs

On the independence number of traceable 2-connected claw-free graphs

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erdős proved that if a graph G with n vertices satisfies e ( G ) > max n − k 2 + k 2 , ⌈ ( n + 1 ) ∕ 2 ⌉ 2 + n − 1 2 2 , where the minimum degree δ ( G ) ≥ k and 1 ≤ k ≤ ( n − 1 ) ∕ 2 , then it is Hamiltonian. For n ≥ 2 k + 1 , let E n k = K k ∨ ( k K 1 + K n − 2 k ) , where “ ∨ ” is the “join” operation. One can observe e ( E n k ) = n − k 2 + k 2 and E n k is not Hamiltonian. As E n k 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.

Pairs of forbidden subgraphs and 2-connected supereulerian graphs

Abstract Let G be a 2-connected claw-free graph. We show that • if G is N 1 , 1 , 4 -free or N 1 , 2 , 2 -free or Z 5 -free or P 8 -free, respectively, then G has a spanning Eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ ( G ) ≥ 3 and G is N 2 , 2 , 5 -free or Z 9 -free, respectively, then G has a spanning Eulerian subgraph or its closure is the line graph of a graph in a family of well-defined graphs. Here Z i ( N i , j , k ) denotes the graph obtained by attaching a path of length i ≥ 1 (three vertex-disjoint paths of lengths i , j , k ≥ 1 , respectively) to a triangle. Combining our results with a result in [Xiong (2014)], we prove that all 2-connected hourglass-free claw-free graphs G with one of the same forbidden subgraphs above (or additionally δ ( G ) ≥ 3 ) are hamiltonian with the same excluded families of graphs. In particular, we prove that every 3-edge-connected claw-free hourglass-free graph that is N 2 , 2 , 5 -free or Z 9 -free is hamiltonian.

Forbidden set of induced subgraphs for [formula omitted]-connected supereulerian graphs

Let H = { H 1 , … , H k } be a set of connected graphs. A graph is said to be H -free if it does not contain any member of H as an induced subgraph. We show that if the following statements hold, • | H | ≥ 2 , • K 1 , 3 ∈ H , • for any integer n 0 , every 2 -connected H -free graph G of order at least n 0 is supereulerian, i . e . , G has a spanning closed trail, then H ∖ { K 1 , 3 } contains an N i , j , k or a path where N i , j , k denotes the graph obtained by attaching three vertex-disjoint paths of lengths i , j , k ≥ 0 to a triangle. As an application, we characterize all the forbidden triples H with K 1 , 3 ∈ H such that every 2 -connected H -free graph is supereulerian. As a byproduct, we also characterize minimal 2-connected non-supereulerian claw-free graphs.

Circumferences of 3-connected claw-free graphs, II

For a graph H , the circumference of H , denoted by c ( H ) , is the length of a longest cycle in H . It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ ≥ 8 , then c ( H ) ≥ min { 9 δ − 3 , n } . In Li (2006), Li conjectured that every 3-connected k -regular claw-free graph H of order n has c ( H ) ≥ min { 10 k − 4 , n } . Later, Li posed an open problem in Li (2008): how long is the best possible circumference for a 3-connected regular claw-free graph? In this paper, we study the circumference of 3-connected claw-free graphs without the restriction on regularity and provide a solution to the conjecture and the open problem above. We determine five families F i ( 1 ≤ i ≤ 5 ) of 3-connected claw-free graphs which are characterized by graphs contractible to the Petersen graph and show that if H is a 3-connected claw-free graph of order n with δ ≥ 16 , then one of the following holds: (a) either c ( H ) ≥ min { 10 δ − 3 , n } or H ∈ F 1 . (b) either c ( H ) ≥ min { 11 δ − 7 , n } or H ∈ F 1 ∪ F 2 . (c) either c ( H ) ≥ min { 11 δ − 3 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 . (d) either c ( H ) ≥ min { 12 δ − 10 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 . (e) if δ ≥ 23 then either c ( H ) ≥ min { 12 δ − 7 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 ∪ F 5 . This is also an improvement of the prior results in Chen (2016), Lai et al. (2016), Li et al. (2009) and Mathews and Sumner (1985).

Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs

A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of $Z_1$ (the graph obtained from a triangle by adding one pendant edge). Our main result in this paper is a characterization of all graphs $H$ such that a 2-connected claw-free graph $G$ is hamiltonian if each end-vertex of every induced copy of $H$ in $G$ has degree at least $|V(G)|/3+1$. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.

Minimum degree conditions for the Hamiltonicity of 3-connected claw-free graphs

Settling a conjecture of Kuipers and Veldman posted in Favaron and Fraisse (2001) [9] , Lai et al. (2006) [15] proved that if H is a 3-connected claw-free simple graph of order n ≥ 196 , and if δ ( H ) ≥ n + 5 10 , then either H is Hamiltonian, or the Ryjáček's closure c l ( H ) = L ( G ) where G is the graph obtained from the Petersen graph P by adding n − 15 10 pendant edges at each vertex of P . Recently, Li (2013) [17] improved this result for 3-connected claw-free graphs H with δ ( H ) ≥ n + 34 12 and conjectured that similar result would also hold even if δ ( H ) ≥ n + 12 13 . In this paper, we show that for any given integer p > 0 and real number ϵ , there exist an integer N = N ( p , ϵ ) > 0 and a family Q ( p ) , which can be generated by a finite number of graphs with order at most max ⁡ { 12 , 3 p − 5 } such that for any 3-connected claw-free graph H of order n > N and with δ ( H ) ≥ n + ϵ p , H is Hamiltonian if and only if H ∉ Q ( p ) . As applications, we improve both results in Lai et al. (2006) [15] and in Li (2013) [17] , and give a counterexample to the conjecture in Li (2013) [17] .