Gyárfás–Sumner conjecture explained
and
complete graph
, the graphs with neither
nor
as
induced subgraphs can be
properly colored using only a constant number of colors. Equivalently, it asks whether the
-free graphs are
-bounded.It is named after
András Gyárfás and
David Sumner, who formulated it independently in 1975 and 1981 respectively. It remains unproven.
In this conjecture, it is not possible to replace
by a graph with cycles. As
Paul Erdős and
András Hajnal have shown, there exist graphs with arbitrarily large chromatic number and, at the same time, arbitrarily large
girth. Using these graphs, one can obtain graphs that avoid any fixed choice of a cyclic graph and clique (of more than two vertices) as induced subgraphs, and exceed any fixed bound on the chromatic number.
The conjecture is known to be true for certain special choices of
, including
paths,
stars, and trees of radius two.It is also known that, for any tree
, the graphs that do not contain any
subdivision of
are
-bounded.
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