Gyárfás–Sumner conjecture

Template:Unsolved In graph theory, the Gyárfás–Sumner conjecture asks whether, for every tree $T$ and complete graph $K$, the graphs with neither $T$ nor $K$ as induced subgraphs can be properly colored using only a constant number of colors. Equivalently, it asks whether the $T$-free graphs are $\chi$-bounded.Template:R It is named after András Gyárfás and David Sumner, who formulated it independently in 1975 and 1981 respectively.Template:R It remains unproven.Template:R

In this conjecture, it is not possible to replace $T$ 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.Template:R 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.Template:R

The conjecture is known to be true for certain special choices of $T$, including paths,Template:R stars, and trees of radius two.Template:R It is also known that, for any tree $T$, the graphs that do not contain any subdivision of $T$ are $\chi$-bounded.Template:R

External links

• Graphs with a forbidden induced tree are chi-bounded, Open Problem Garden