## Abstract Let __G__ be a graph on __n__ vertices and __N__~2~(__G__) denote the minimum size of __N__(__u__) βͺ __N__(__v__) taken over all pairs of independent vertices __u, v__ of __G__. We show that if __G__ is 3βconnected and __N__~2~(__G__) β©Ύ Β½(__n__ + 1), then __G__ has a Hamilton cycle. We
Neighborhood unions and regular factors
β Scribed by Thomas Niessen
- Publisher
- John Wiley and Sons
- Year
- 1995
- Tongue
- English
- Weight
- 737 KB
- Volume
- 19
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
We examine bounds on the size of the neighborhood union for two (independent) vertices of a graph that imply the existence of regular factors.
π SIMILAR VOLUMES
Let G be a simple graph of order n with connectivity k 3 2, independence number cc We prove that if for each independent set S of cardinality k+ 1, one of the following condition holds: (1) there exist u # v in S such that d(u) +d(v) > n or ) N(u)nN(v) I> cr; (2) for any distinct pair u and u in S,
Let G be a graph of order n. In this paper, we prove that if G is a 2-connected graph of order n such that for all u, ve V(G), 2 where dist(u,v) is the distance between u and v in G, then either G is hamiltonian, or G is a spanning subgraph of a graph in one of three families of exceptional graphs.
We examine several extremal problems for graphs satisfying the property JN(x) u N(y)l ? s f o r every pair of nonadjacent vertices x , y E V ( G ) . In particular, values for s are found that ensure that the graph contains an s-matching, a 1-factor, a path of a specific length, or a cycle of a spec
## Abstract Dirac proved that a graph __G__ is hamiltonian if the minimum degree $\delta(G) \geq n/2$, where __n__ is the order of __G__. Let __G__ be a graph and $A \subseteq V(G)$. The neighborhood of __A__ is $N(A)=\{ b: ab \in E(G)$ for some $a \in A\}$. For any positive integer __k__, we show