In this paper we obtain two sufficient conditions, Ore type (Theorem 1) and Dirac type (Theorem 2). on the degrees of a bipartite oriented graph for ensuring the existence of long paths and cycles. These conditions are shown to be the best possible in a sense. An oriented graph is a digraph without
Longest paths and cycles in K1,3-free graphs
β Scribed by Manton M. Matthews; David P. Sumner
- Publisher
- John Wiley and Sons
- Year
- 1985
- Tongue
- English
- Weight
- 383 KB
- Volume
- 9
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
In this article w e show that the standard results concerning longest paths and cycles in graphs can be improved for K,,,-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K,,,-free graphs.
π SIMILAR VOLUMES
## Abstract For a graph __G__, let __p(G)__ denote the order of a longest path in __G__ and __c(G)__ the order of a longest cycle in __G__, respectively. We show that if __G__ is a 3βconnected graph of order __n__ such that $\textstyle{\sum^{4}\_{i=1}\,{\rm deg}\_{G}\,x\_{i} \ge {3\over2}\,n + 1}$
In this article, we establish bounds for the length of a longest cycle C in a 2-connected graph G in terms of the minimum degree Ξ΄ and the toughness t. It is shown that C is a Hamiltonian cycle or |C| β₯ (t + 1)Ξ΄ + t.
## Abstract We show that every connected __K__~1,3~βfree graph with minimum degree at least __2k__ contains a __k__βfactor and construct connected __K__~1,3~βfree graphs with minimum degree __k__ + __0__(β__k__) that have no __k__βfactor.
There have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K,,,-free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K,.3-free gra