We present and prove several results concerning the length of longest cycles in 2connected or I-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings.
Long cycles in graphs with large degree sums
โ Scribed by Douglas Bauer; H.J. Veldman; A. Morgana; E.F. Schmeichel
- Publisher
- Elsevier Science
- Year
- 1990
- Tongue
- English
- Weight
- 764 KB
- Volume
- 79
- Category
- Article
- ISSN
- 0012-365X
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๐ SIMILAR VOLUMES
## Abstract For a graph __G, p__(__G__) denotes the order of a longest path in __G__ and __c__(__G__) the order of a longest cycle. We show that if __G__ is a connected graph __n__ โฅ 3 vertices such that __d__(__u__) + __d__(__v__) + __d__(__w__) โง n for all triples __u, v, w__ of independent verti
## Abstract Let __G__ be a simple graph of order __n__ and minimal degree >โcn (0โ<โcโ<โ1/2). We prove that (1) There exist __n__~0~โ=โ__n__~0~(__c__) and __k__โ=โ__k__(__c__) such that if __n__โ>โ__n__~0~ and __G__ contains a cycle __C__~__t__~ for some __t__โ>โ2__k__, then __G__ contains a cycle
## Abstract Our main result is the following theorem. Let __k__โโฅโ2 be an integer, __G__ be a graph of sufficiently large order __n__, and __ฮด__(__G__)โโฅโ__n__/__k__. Then: __G__ contains a cycle of length __t__ for every even integer __t__โโโ[4, __ฮด__(__G__)โ+โ1]. If __G__ is nonbipartite then