$\chi$-bounds, operations and chords
  A \emph{long unichord} in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is \emph{long-unichord-free} if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free graph. We give an $O(n^4m)$-time algorithm to recognize them. We show that any long-unichord-free graph $G$ can be colored with at most $O(\omega^3)$ colors, where $\omega$ is the maximum number of pairwise adjacent vertices in $G$. 
Lan Anh Pham and Nicolas Trotignon
cs.DM cs.DS math.CO