The Gyárfás-Sumner conjecture asserts that for each fixed tree 
, the class of graphs with
no induced subgraph isomorphic to 
is a chi-bounded graph
class (Sumner 1981, Gyárfás 1987). Equivalently, for every positive
integer 
there is a bound depending only on 
and 
on the chromatic number
of any such graph with clique
number at most 
.
The conjecture holds when 
is a path graph, a star
graph, or a tree of graph
radius at most 2 (Scott and Seymour 2020). The conjecture
is equivalent to the corresponding statement for every forest,
since every forest is an induced
subgraph of a tree.
-Boundedness." J. Graph Th. 95, 473-504,
2020.