4 problems

Let $G$ be a graph, let $k(G)$ be its competition number, and let $h(G)$ be the number of holes of $G$, where a hole is a cycle of length at least $4$ occurring as an induced subgr…

Competition-number equality conjecture. Every graph $G$ satisfies

123-avoidance conjecture.

Let $G$ be a graph. Its hole space is the vector space generated by the holes of $G$ in the cycle space of $G$, and let $k(G)$ denote its competition number. Hole-space dimension c…