Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and $S\subseteq V(G)$ takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S.

We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows:

Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.

Hadwiger 著名的着色猜想指出，每个t色图都包含一个$K_t$-次要的。Holroyd [11]推测了以下对 Hadwiger 猜想的强化：如果G是t色图并且  $S\subseteq V（G）$获取G的每个t着色中的所有颜色，则G包含$K_t$-次要根于 S。

我们在第一个公开案例中证明了这个猜想$t=4$。值得注意的是，我们的结果还直接暗示了 Hadwiger 对 5 色图的猜想的更强版本，如下所示：

每个 5 色图都包含一个$K_5$-minor 具有单例分支集。事实上，在 5 顶点关键图中，我们可以将单例分支集指定为图的任何顶点。