Erdős–Lovász double-critical graph conjecture

Let $G$ be a connected graph with chromatic number $t$. It is double-critical if for every edge $xy\in E(G)$, the graph $G\backslash\{x,y\}$ is $(t-2)$-colorable. The complete graphs are double-critical. Erdős–Lovász double-critical graph conjecture. If $G$ is a double-critical, $t$-chromatic graph, then

$$G=K_t.$$

Complete graphs are the only known examples of double-critical graphs. The conjecture has been verified for $t\le 5$ and remains open for $t\ge 6$.