Graphs with no even holes and no sector wheels are the union of two chordal graphs

Abstract

Sivaraman conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1, X_2 \subseteq V(G)$ satisfying $V(G) = X_1 \cup X_2$ such that the induced graphs $G[X_1]$ and $G[X_2]$ are both chordal. We prove this conjecture in the special case where $G$ contains no sector wheel, namely, a pair $(H, w)$ where $H$ is an induced cycle of $G$ and $w$ is a vertex in $V(G) \setminus V(H)$ such that $N(w) \cap H$ is either $V(H)$ or a path with at least three vertices.