关于图因子的邻集条件

设F为图G的一个支撑子图.如果对所有x∈V(G),有d_F(x)∈{1,3,…,2n-1),则称F为G的一个(1,3,…,2n-1)一因子;如果对所有x∈V(G),有d_F(x)=k,则称F为G的一个k-因子.本文以图的顶点邻集对一个图具有包含任一条给定边的{1,3,…,2n-1)-因子和k-因子分别给出了充分条件.     

Uniquely Hamiltonian Graphs of Minimum Degree 4

We construct an infinite family of uniquely hamiltonian graphs of minimum degree 4, maximum degree 14, and of arbitrarily high maximum degree.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.     

半无爪可迹图的度和条件

可迹图即为一个含有Hamilton路的图.令$N[v]=N(v)\cup\{v\}$, $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$.若图中任意距离为2的两点$u,v$满足$J(u,v)\neq \emptyset$,则称该图为半无爪图.令$\sigma_{k}(G)=\min\{\sum_{v\in S}d(v):S$为$G$中含有$k$个点的独立集\},其中$d(v)$表示图$G$中顶点$v$的度.本论文证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{3}(G)\geq {n-2}$,则图$G$为可迹图; 文中给出一个图例,说明上述结果中的界是下确界; 此外,我们证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{2}(G)\geq \frac{2({n-2})}{3}$,则该图为可迹图.     

P3-支配图哈密尔顿性的两个充分条件

在文献[3]中介绍了一个新的图类-P3-支配图.这个图类包含所有的拟无爪图,因此也包含所有的无爪图.在本文中,我们证明了每一个点数至少是3的三角形连通的P3-支配图是哈密尔顿的,但有一个例外图K1,1,3.同时,我们也证明了k-连通的(k≥2)的P3-支配图是哈密尔顿的,如果an(G)≤k,但有两个例外图K1,1,3 and K2,3.     

Degree Sum Conditions for Hamiltonicity on k-Partite Graphs

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graph G has order p and minimum degree at least \(\frac{p} {2}\) then G is hamiltonian. Moon and Moser showed that if G is a balanced bipartite graph (the two partite sets have the same order) with minimum degree more than \(\frac{p} {4}\) then G is hamiltonian. Recently their idea is generalized to k-partite graphs by Chen, Faudree, Gould, Jacobson, and Lesniak in terms of minimum degrees. In this paper, we generalize this result in terms of degree sum and the following result is obtained: Let G be a balanced k-partite graph with order kn. If for every pair of nonadjacent vertices u and v which are in different parts $$d(u) + d(v) > \left\{ {\begin{array}{*{20}c} {\left( {k - \frac{2} {{k + 1}}} \right)n} & {if k is odd} \\ {\left( {k - \frac{4} {{k + 2}}} \right)n} & {if k is even} \\ \end{array} } \right.,$$ then G is hamiltonian.     

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d _G (v) be the degree of a vertex v in a graph G. For G[X, Y] and ${S \subseteq V(G),}$ we define ${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$ . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ _1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and ${S \subseteq V(G)}$ such that σ _1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or ${|S \cap X| > |Y|}$ and there exists a cycle containing Y. This degree sum condition is sharp.     

Neighborhood Union Conditions for Hamiltonicity of P 3-Dominated Graphs

As an extension of quasi claw-free graphs, the class of P ₃-dominated graphs has been introduced by Broersma and Vumar (Math Methods Oper Res 69:297–306, 2009). For a noncomplete graph G, the number NC and NC ₂ are defined as \({NC=\min\{|N(x)\cup N(y)|: x,y\in V(G) {\rm and} xy\notin E(G)\}\, {\rm and} NC_2=\min\{|N(x)\cup N(y)|: x,y\in V(G)\, {\rm and}\, d(x,y)=2 \}}\) , respectively. For a complete graph G, set \({NC=NC_{2}=|V(G)|-1}\) . In this paper, we prove that a 2-connected P ₃-dominated graph of order n is traceable if \({NC\geq (n-2)/2}\) . Moreover, we prove that a 3-connected P ₃-dominated graph of order n is hamiltonian if \({NC_2\geq (2n-6)/3}\) . Our results extend some previous results on claw-free graphs.     

Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G.     

Hamiltonian Kneser Graphs

The Kneser graph K(n,k) has as vertices the k-subsets of {1, 2, ..., n}. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the first author [2] that Kneser graphs have Hamilton cycles for n >= 3k. In this note, we give a short proof for the case when k divides n. Received September 14, 1999     

Induced-Universal Graphs for Graphs with Bounded Maximum Degree

For a family \(\mathcal {F}\) of graphs, a graph U is induced-universal for \({\mathcal{F}}\) if every graph in \({\mathcal{F}}\) is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most r, which has \(Cn^{\lfloor (r+1)/2\rfloor}\) vertices and \(Dn^{2\lfloor (r+1)/2\rfloor -1}\) edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such an induced-universal graph must have at least cn ^r/2 vertices for some c depending only on r.Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. The construction also extends to multigraphs and directed graphs with bounded degree.     

Degree-Light-Free Graphs and Hamiltonian Cycles

 Let G and H be graphs. G is said to be degree-light H-free if G is either H-free or, for every induced subgraph K of G with K≅H, and every {u,v}⊆K, d i s t _K (u,v)=2 implies max {d(u),d(v)}≥|V(G)|/2. In this paper, we will show that every 2-connected graph with either degree-light {K _1,3, P ₆}-free or degree-light {K _1,3, Z}-free is hamiltonian (with three exceptional graphs), where P ₆ is a path of order 6 and Z is obtained from P ₆ by adding an edge between the first and the third vertex of P ₆ (see Figure 1). Received: December 9, 1998?Final version received: July 21, 1999     

Hamiltonian Connected Claw-Free Graphs

A well-known result by O. Ore is that every graph of order n with d(u)+d(v)n+1 for any pair of nonadjacent vertices u and v is hamiltonian connected (i.e., for every pair of vertices, there is a hamiltonian path joining them). In this paper, we show that every 3-connected claw-free graph G on at most 5–8 vertices is hamiltonian connected, where denotes the minimum degree in G. This result generalizes several previous results.Acknowledgments. The author would like to thank the referee for his important suggestions and careful corrections.Final version received: March 12, 2003Supported by the project of Nature Science Funds of China     

门槛图与度极大图

证明了门槛图与度极大图是一类图的两种不同说法,同时用图的对角限制极左矩阵刻画这一类图的结构。     

A Note on Degree Conditions for Traceability in Locally Claw-Free Graphs

We say that a set S of vertices is traceable in a graph G whenever there is a path in G containing all vertices of S. In this paper we study the problem of traceability of a prescribed set of vertices in a locally claw-free graph (i.e. a graph in which some specified vertices are not centers of an induced claw). In particular we give sufficient degree conditions restricted to the given set S of vertices for the traceability of S.     

一个关于图是[a,b;n]-均匀的度条件

设t,a,b和n为整数且1≤a＜b,t≥3以及n≥1.如果G的导出子图不含有K1,t,则该图G称为K1,t-无爪图.如果对于图G中含有n条边的任意匹配M,都在G中有[a,b]-因子F包含M以及在G中有另一个[a,b]-因子F'不包含M,则图G称为[a,b;n]-均匀图.给出了K1,t-无星图G是[a,b;n]-均匀图的度条件.进一步,指出本文中的结果在某种意义上说是最佳的.     

Hamiltonian图和弱Ore条件

本文用弱 Ore条件和 NC2去研究 Hamiltonian图 ,得到深刻的结果 ,它们改进和统一文献 [1 ]至 [5 ]中熟知的一些著名结果 .