For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$ if $\chi(G) \leq f(\omega(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a quadratic $\chi$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork, $F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $\chi(G)\le \omega(G)^2$. We also note that the class $\cal G$ does not admit a linear $\chi$-binding function.
The claw is the graph $K_{1,3}$, and the fork is the graph obtained from the claw $K_{1,3}$ by subdividing one of its edges once. In this paper, we prove a structure theorem for the class of (claw, $C_4$)-free graphs that are not quasi-line graphs, and a structure theorem for the class of (fork, $C_4$)-free graphs that uses the class of (claw, $C_4$)-free graphs as a basic class. Finally, we show that every (fork, $C_4$)-free graph $G$ satisfies $\chi(G)\leqslant \lceil\frac{3\omega(G)}{2}\rceil$ via these structure theorems with some additional work on coloring basic classes.
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