Exact zero-sum Ramsey numbers for friendship graphs, books, and triangular windmills

We determine exact zero-sum Ramsey numbers modulo three for friendship graphs,
books, and triangular windmills with pendant leaves, and we prove a bound for connected cactus graphs. If $F_t$ is the friendship graph formed
by $t$ triangles with one common vertex, then
\[
 R(F_t,\mathbb{Z}_3)=
 \begin{cases}
  11,&t=1,\\
  9,&t=2,\\
  2t+2,&t\ge3\text{ and }3\mid t,\\
  2t+3,&t\ge4\text{ and }3\nmid t.
 \end{cases}
\]
If $W_{\ell,t}$ is obtained by adding $\ell$ pendant edges at the common
center of $F_t$, then for $\ell\ge3$ divisible by three,
\[
 R(W_{\ell,t},\mathbb{Z}_3)=
 \begin{cases}
 |V(W_{\ell,t})|,&3\mid t,\\
 |V(W_{\ell,t})|+1,&3\nmid t.
 \end{cases}
\]
If $B_t$ is the book formed by $t$ triangles with one common edge, then
$R(B_1,\mathbb{Z}_3)=11$ and $R(B_t,\mathbb{Z}_3)=t+4$ for every $t\ge4$ with
$t\equiv1\pmod3$. Finally, among connected cacti whose number of edges is
divisible by three, $K_3$ and $F_2$ are the only ones with
$R(G,\mathbb{Z}_3)>|V(G)|+3$.