Gyárfás–Sumner conjecture

Unsolved problem in mathematics: Does forbidding both a tree and a clique as induced subgraphs produce graphs of bounded chromatic number? (more unsolved problems in mathematics)

In graph theory, the Gyárfás–Sumner conjecture says that for any fixed pair of a tree $T$ and a complete graph $K_t$, every graph that is both $T$-free and $K_t$-free is also $\chi$-bounded; that is, every graph that contains neither $T$ nor $K_t$ as an induced subgraph can be properly colored using a constant number of colors. Equivalently, it says that every $K_t$-free graph $G$ contains any arbitrarily chosen tree $T$ (as an induced subgraph), as long as $\chi (G)$ is large enough. 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 $T$ 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. 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 $T$, including paths, stars, and trees of radius two. It is also known that, for any tree $T$, the graphs that do not contain any subdivision of $T$ are $\chi$-bounded.

External links

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