# Even and odd holes in cap-free graphs

@article{Conforti1999EvenAO, title={Even and odd holes in cap-free graphs}, author={Michele Conforti and G{\'e}rard Cornu{\'e}jols and Ajai Kapoor and Kristina Vuskovic}, journal={J. Graph Theory}, year={1999}, volume={30}, pages={289-308} }

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is…

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## 51 Citations

On the chromatic number of a family of odd hole free graphs

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A class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width is presented, which gives a negative answer and cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem.

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