Fan-type Condition Research Articles

Disjoint cycles and 2-factors with Fan-type condition in a graph

Let k be a positive integer. For a graph G, let σ12(G) be the minimum value of max{d(x),d(y)} for any pair of nonadjacent vertices x and y. Corrádi and Hajnal proved that every graph G of order n≥3k with δ(G)≥2k contains k disjoint cycles. In this paper, we generalize this theorem by proving that, for a graph G of order at least 3k+2, if σ12(G)≥2k, then G contains k disjoint cycles. The bound of order is sharp. Based on this result, we show that if G is a 2-connected graph of order at least 3k+5 with σ12(G)≥max{n/2,2k+2}, then G contains a 2-factor with exactly k disjoint cycles, which improves the theorem of Brandt et al. (1997) [2].

A Fan-type condition for graphs to be [formula omitted]-leaf-connected

For k≥2, a graph G is said to be k-leaf connected if |G|>k and for each subset S of V(G) with |S|=k, G has a spanning tree T with precisely S as the set of endvertices of T. This property is a general concept of Hamiltonian-connected. This paper gives a Fan-type condition for graphs to be k-leaf-connected.

Long Paths in Bipartite Graphs and Path-Bistar Bipartite Ramsey Numbers

In this paper, we focus on a so-called Fan-type condition assuring us the existence of long paths in bipartite graphs. As a consequence of our main result, we completely determine the bipartite Ramsey numbers $$b(P_{s},B_{t_{1},t_{2}})$$, where $$B_{t_{1},t_{2}}$$ is the graph obtained from a $$t_{1}$$-star and a $$t_{2}$$-star by joining their centers.

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.

Fan-type condition on disjoint cycles in a graph

Let k be a positive integer. In this paper, we prove that for a graph G with at least 4k vertices, if max{d(x),d(y)}≥2k for any pair of nonadjacent vertices {x,y}⊆V(G), then G contains k disjoint cycles. This generalizes the results given by Corrá di and Hajnal (1963), Enomoto (1998), and Wang (1999).

Fan-Type Conditions for Spanning Eulerian Subgraphs

For a graph $$G$$G, let $$\delta _F(G)=\min \{\max \{d(u), d(v)\} | \text{ for } \text{ any }~u, v\in V(G)\, \text{ with } \text{ distance }~2\}$$?F(G)=min{max{d(u),d(v)}|foranyu,v?V(G)withdistance2}. A graph is supereulerian if it has a spanning Eulerian subgraph. Let $$p>0$$p>0, $$g>2$$g>2 and $$\epsilon $$∈ be given nonnegative numbers. Let $$\mathcal{Q}$$Q be the family of non-supereulerian graphs with order at most $$5(p-2)$$5(p-2). In this paper, we prove that for a 3-edge-connected graph $$G$$G of order $$n$$n, if $$G$$G satisfies a Fan-type condition $$\delta _F(G)\ge \frac{n}{(g-2)p}-\epsilon $$?F(G)?n(g-2)p-∈ and $$n$$n is sufficiently large, then $$G$$G is supereulerian if and only if $$G$$G is not contractible to a graph in $$\mathcal{Q}$$Q. Results on best possible values of $$p$$p and $$\epsilon $$∈ for such graphs to either be supereulerian or be contractible to the Petersen graph are given.

Fan type condition and characterization of Hamiltonian graphs

Fan type condition and characterization of Hamiltonian graphs

Fan-type results for the existence of [formula omitted]-factors

Let 1 ≤ a < b be integers and G a graph of order n sufficiently large for a and b. Then G has an [ a , b ] -factor if the minimum degree is at least a and every pair of vertices distance two apart has cardinality of the neighborhood union at least an / ( a + b ) . This lower bound is sharp. As a consequence, we have a Fan-type condition for a graph to have an [ a , b ] -factor.

A fan-type condition for cyclability

A graph G is said to be cyclable if for each orientation D of G, there exists a set S ( D ) ⊆ V ( G ) such that reversing all the arcs with one end in S results in a hamiltonian digraph. Let G be 4-connected simple graph of even order n ⩾ 14 . In this paper, we show that if max { d ( u ) , d ( v ) } ⩾ ( n + 1 ) / 2 for any u , v ∈ V ( G ) with d ( u , v ) = 2 , then G is cyclable.

A Fan Type Condition For Heavy Cycles in Weighted Graphs

A weighted graph is a graph in which each edge $e$ is assigned a non-negative number $w(e)$, called the weight of $e$. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex $v$ is the sum of the weights of the edges incident with $v$. In this paper, we prove the following result: Suppose $G$ is a 2-connected weighted graph which satisfies the following conditions: 1. $\max\{d^w(x),d^w(y)\mid d(x,y)=2\}\geq c/2$; 2. $w(xz)=w(yz)$ for every vertex $z\in N(x)\cap N(y)$ with $d(x,y)=2$; 3. In every triangle $T$ of $G$, either all edges of $T$ have different weights or all edges of $T$ have the same weight. Then $G$ contains either a Hamilton cycle or a cycle of weight at least $c$. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result.

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.