### Restricted size Ramsey number for P3 versus cycle

#### Abstract

*F*,

*G*and

*H*be simple graphs. We say

*F*→ (

*G*,

*H*) if for every 2-coloring of the edges of

*F*there exists a red copy of

*G*or a blue copy of

*H*in

*F*. The Ramsey number

*r*(

*G,H*) is defined as

*r*(

*G,H*) = min{|

*V*(

*F*)|:

*F*→ (

*G,H*)}, while the restricted size Ramsey number

*r*

^{*}(

*G,H*) is defined as

*r*

^{*}(

*G,H*) = min{|

*E*(

*F*)|:

*F*→ (

*G,H*),|

*V*(

*F*)| =

*r*(

*G,H*)}. In this paper we determine previously unknown restricted size Ramsey numbers

*r*

^{*}(

*P*

_{3},

*C*) for 7 ≤

_{n}*n*≤ 12. We also give new upper bound

*r*

^{*}(

*P*

_{3},

*C*

_{n}) ≤ 2

*n*-2 for

*n*≥ 10 and

*n*is even.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.12

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