On the resilience of long cycles in random graphs.

Dellamonica, Domingos jun.; Kohayakawa, Yoshiharu; Marciniszyn, Martin; Steger, Angelika

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A note on the Size-Ramsey number of long subdivisions of graphs

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Let $T_{s} H$ be the graph obtained from a given graph $H$ by subdividing each edge $s$ times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph $H$ , there exist graphs $G$ with $O (s)$ edges that are Ramsey with respect to $T_{s} H$ .