Ramsey numbers for the pair sparse graph-path or cycle

Let $G$ be a connected graph on $n$ vertices with no more than $n(1 + \varepsilon )$ edges, and ${P_k}$ or ${C_k}$ a path or cycle with $k$ vertices. In this paper we will show that if $n$ is sufficiently large and $\varepsilon$ is sufficiently small then for $k$ odd

$$r(G,\,{C_k}) = 2n - 1.$$

Also, for $k \geqslant 2$,

$$r(G,\,{P_k}) = \max \{ n + [k/2] - 1,\,n + k - 2 - \alpha \prime - \delta \} ,$$

where $\alpha \prime$ is the independence number of an appropriate subgraph of $G$ and $\delta$ is 0 or $1$ depending upon $n$, $k$ and $\alpha \prime$.