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Large Monotone Paths in Graphs with Bounded Degree

โœ Scribed by Raphael Yuster


Publisher
Springer Japan
Year
2001
Tongue
English
Weight
109 KB
Volume
17
Category
Article
ISSN
0911-0119

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## 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.

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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\

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## 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__ โ‰ฅ 3 vertices such that __d__(__u__) + __d__(__v__) + __d__(__w__) โ‰ง n for all triples __u, v, w__ of independent verti