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For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denotedby $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used bythe 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 afunction $f(x)$ with $f(x)\rightarrow \infty$ as $x\rightarrow \infty$ suchthat for any graph $G$ from the family one has $\chi_{_{\sf FF}}(G)\geqf(\delta(G))$, where $\delta(G)$ is the minimum degree of $G$. We first givesome results concerning $(\delta, \chi_{_{\sf FF}})$-bounded families andobtain 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 thevalidity of this conjecture for chordal graphs, complement of bipartite graphsand graphs with low minimum degree.

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