Restricted size Ramsey number for $P_3$ versus cycles

TL;DR

This paper determines the restricted size Ramsey numbers for paths versus cycles for specific cases and provides new bounds, advancing understanding in graph Ramsey theory.

Contribution

It computes previously unknown restricted size Ramsey numbers for $P_3$ versus cycles and establishes a new upper bound for even cycles.

Findings

Calculated $r^{*}(P_3, C_n)$ for $7 \,\leq n \leq 12$

Established $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$

Enhanced bounds in graph Ramsey theory

Abstract

Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F \rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ is defined as $r^{*}(G, H) = min\{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)\}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$.