Hamiltonian graphs involving distances

Let G be a graph of order n. We show that if G is a 2-connected graph and max{d(u), d(v)} + |N(u) U N(v)| ≥ n for each pair of vertices u, v at distance two, then either G is hamiltonian or G ?3K_n/3 U T₁ U T₂, where n ? O (mod 3), and T₁ and T₂ are the edge sets of two vertex disjoint triangles containing exactly one vertex from each K_n/3. This result generalizes both Fan's and Lindquester's results as well as several others.     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

One sufficient condition for hamiltonian graphs

Let G be a 2-connected graph of order n. We show that if for each pair of nonadjacent vertices x,y ∈ V(G), then G is Hamiltonian.     

Hamiltonian graphs involving neighborhood unions

Dirac proved that a graph G is hamiltonian if the minimum degree , where n is the order of G. Let G be a graph and . The neighborhood of A is for some . For any positive integer k, we show that every (2k ? 1)‐connected graph of order n ≥ 16k³ is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k ? 1 on connectivity is best possible in general while the lower bound 16k³ for n is conjectured to be unnecessary. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 83–100, 2006     

路可扩图的一个充分条件

通过讨论图中任意一对不相邻顶点的度和,对路可扩图的充分条件进行研究,得到了如下结果：设图G的阶是n,如果G中任意一对不相邻顶点的度和至少为3/2n-1,则图G是路可扩的.并且说明了这里两不相邻顶点的度和的下界3/2n-1是最好可能的.     

A new sufficient condition for hamiltonian graphs

We prove that a k-connected graph of order n, such that the number of neighbors of every independent set of k vertices is greater than (k(n – 1))/(k + 1) is hamiltonian.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

A sufficient condition for graphs to be weaklyk-linked

We consider graphs, which are finite, undirected, without loops and in which multiple edges are possible. For each natural numberk letg(k) be the smallest natural numbern, so that the following holds:LetG be ann-edge-connected graph and lets ₁,...,s _k,t ₁,...,t _k be vertices ofG. Then for everyi {1,..., k} there existsa pathP _i froms _i tot _i, so thatP ₁,...,P _k are pairwise edge-disjoint. We prove      

A sufficient condition for boundedness of tolerance graphs

Golumbic, Monma, and Trotter showed that every tolerance graph for which no vertex neighborhood is contained in another vertex neighborhood is a bounded tolerance graph. We strengthen this result by weakening the neighborhood condition. In this way, more tolerance graphs can be recognized as bounded. Our argument relies on a variation of the concept of “assertive vertices”.     

A sufficient condition for Hamiltonian cycles in bipartite tournaments

In this paper, we present a new sufficient condition on degrees for a bipartite tournament to be Hamiltonian, that is, if an n × n bipartite tournament T satisfies the condition W(n - 3), then T is Hamiltonian, except for four exceptional graphs. This result is shown to be best possible in a sense.     

A new neighborhood union condition for Hamiltonian graphs

For a vertex set {u ₁,u ₂,...,u _k} of a graphG withn vertices, let $$\begin{gathered} s(G;\{ u_1 ,u_2 ,...,u_k \} ) = \sum {1 \leqslant i< j \leqslant k\left| {N(u_i ) \cup N(u_j )} \right|,} \hfill \\ NC_k = \min \{ s(G;\{ x_1 ,...,x_k \} )\} :\{ x_1 ,...,x_k \} is an independent set\} . \hfill \\ \end{gathered} $$ In this paper, we shall prove that ifG is 3-connected andNC ₄≥3n, thenG is either a hamiltonian or Petersen graph. This generalizes some results on the neighborhood union conditions for hamiltonian graphs.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

A sufficient condition for the liveness of weighted event graphs

Weighted event graphs (in short WEG) are widely used to model industrial problems and embedded systems. In an optimization context, fast algorithms checking the liveness of a marked WEG must be developed. The purpose of this paper is to develop a sufficient condition of liveness of a WEG. We first show that any unitary WEG can be transformed into a graph in which the values of the arcs adjacent to any transition depend on the transition. Then, a simple sufficient condition of liveness can be expressed on this new graph and polynomially computed. This condition is shown to be necessary for a circuit with two transitions.     

Strong sufficient conditions for the existence of Hamiltonian circuits in undirected graphs

In this paper, we derive some results giving sufficient conditions for a graph G containing a Hamiltonian path to be Hamiltonian. In particular the Bondy-Chvátal theorem [J. A. Bondy and V. Chvátal, Discrete Math. 15 (1976), 111–135] is derived as a corollary of the main theorem of this paper and hence a more powerful closure operation than the one introduced by Bondy and Chvátal is defined. These results can be viewed as a step towards a unification of the various known results on the existence of Hamiltonian circuits in undirected graphs. Moreover, Theorem 1 of this paper provides a counterpart of the Chvátal-Erdös theorem [V. Chvátal and P. Erdös, Discrete Math. 2 (1972), 111–113] which gives a sufficient condition for a Hamiltonian circuit in terms of global vertex connectivity and independence number.     

A sufficient condition for bipartite graphs to be type one

The total chromatic number χ_T(G) of graph G is the least number of colors assigned to V(G) ∪ E(G) such that no adjacent or incident elements receive the same color. In this article, we give a sufficient condition for a bipartite graph G to have χ_T(G) = Δ(G) + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 133–137, 1998     

A sufficient condition on upper embeddability of graphs

In this paper we mainly prove that let G be a(k + 1)-edge-connected simple graph of order n with girth g.Then G is upper embeddable if for any independent set I(G) = {vi | 1 i k2 + 2},k = 0,1,2 and the lower bound is tight.