## Abstract Let __ex__ \* (__D__; __H__) denote the maximum number of edges in a connected graph with maximum degree __D__ and no induced subgraph isomorphic to __H.__ We prove that this is finite only when __H__ is a disjoint union of paths,m in which case we provide crude upper and lower bounds.
Large Monotone Paths in Graphs with Bounded Degree
โ Scribed by Raphael Yuster
- Publisher
- Springer Japan
- Year
- 2001
- Tongue
- English
- Weight
- 109 KB
- Volume
- 17
- Category
- Article
- ISSN
- 0911-0119
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
Let ex\*(D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex\*(D;H) = O(D2ex\*(D;Pm)). Constructively, ex\*(D;qPm) = O(qD2ex\
## 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 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