$(\delta, \chi_{_{\sf FF}})$-bounded families of graphs

For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denoted by $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. We call a family $\mathcal{F}$ of graphs $(\delta, \chi_{_{\sf FF}})$-bounded if there exists a function $f(x)$ with $f(x)\rightarrow \infty$ as $x\rightarrow \infty$ such that for any graph $G$ from the family one has $\chi_{_{\sf FF}}(G)\geq f(\delta(G))$, where $\delta(G)$ is the minimum degree of $G$. We first give some results concerning $(\delta, \chi_{_{\sf FF}})$-bounded families and obtain a few such families. Then we prove that for any positive integer $\ell$, $Forb(K_{\ell,\ell})$ is $(\delta, \chi_{_{\sf FF}})$-bounded, where $K_{\ell,\ell}$ is complete bipartite graph. We conjecture that if $G$ is any $C_4$-free graph then $\chi_{_{\sf FF}}(G)\geq \delta(G)+1$. We prove the validity of this conjecture for chordal graphs, complement of bipartite graphs and graphs with low minimum degree.