Consecutive holes
نویسندگان
چکیده
A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for all ν > 0, every triangle-free graph with sufficiently large chromatic number contains holes of ν consecutive lengths. In particular, this implies two well-known conjectures of Gyárfás [3], namely that every triangle-free graph with sufficiently large chromatic number has a hole of length at least k, and every such graph has an odd hole of length at least k. It was not known until now that every such graph has a hole of length > 7.
منابع مشابه
Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes
A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for all ν > 0, every triangle-free graph with sufficiently large chromatic number contains holes of ν consecutive lengths.
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