Clique minors in double-critical graphs

A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erd\H{o}s and Lov\'asz that the complete graphs are the only double-critical $t$-chromatic graphs remains open for all $t\ge6$. Given the difficulty in settling Erd\H{o}s and Lov\'asz's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every double-critical $t$-chromatic graph contains a $K_t$ minor and verified their conjecture for $t\le7$. A computer-assisted proof of their conjecture for $t=8$ was recently announced by Albar and Gon\c{c}alves. In this paper we give a much shorter and computer-free proof of their conjecture for $t\le8$ and prove the next step by showing that every double-critical $t$-chromatic graph contains a $K_9$ minor for all $t\ge9$.