Every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable

Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. We say that a graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable. In this paper we consider the next step and prove that every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable. Our result implies that $H$-Hadwiger's Conjecture, suggested by Paul Seymour in 2017, is true for every graph $H$ on eight vertices such that the complement of $H$ has maximum degree at least four, a perfect matching, a triangle and a cycle of length four. Our proof utilizes an extremal function for $\mathcal{K}_8^{-4}$ minors obtained in this paper, generalized Kempe chains of contraction-critical graphs by Rolek and the second author, and the method for finding $K_7$ minors from three different $K_5$ subgraphs by Kawarabayashi and Toft; this method was first developed by Robertson, Seymour and Thomas in 1993 to prove Hadwiger's Conjecture for $t=6$.