Two subgraphs $A,B$ of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results.

We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number: that is, for all $t,c\ge 1$ there exists $d\ge 1$ such that if G has chromatic number at least d, and does not contain the complete graph $K_t$ as a subgraph, then there are anticomplete subgraphs $A,B$, where A has minimum degree at least c and B has chromatic number at least c.

Second, we look at what happens if we replace the hypothesis that G has sufficiently large chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding $K_t$, is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding $K_t$ we exclude the complete bipartite graph $K_{t,t}$. More exactly: for all $t,c\ge 1$ there exists $d\ge 1$ such that if G has minimum degree at least d, and does not contain the complete bipartite graph $K_{t,t}$ as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.

两个子图$A,乙$如果图G的顶点不相交并且没有边连接它们，则它们是反完备的。如果G是一个有界团数和足够大的色数 的图，那么它是否有两个反完全子图，并且都具有很大的色数？这是 El-Zahar 和 Erdős 于 1986 年提出的问题，目前仍悬而未决。如果是这样，那么至少应该有两个反完全子图，它们的最小度都很大，这就是我们的结果之一。

我们证明了它的两个变体。首先，一个强化：我们可以要求两个子图之一有大的色数：也就是说，对于所有$t,C\ge 1$那里存在$d\ge 1$使得如果G的色数至少为d，并且不包含完整图$K_t$作为子图，则存在反完全子图$A,乙$，其中A的最小阶数至少为c，而B 的色数至少为c。

其次，我们看看如果我们用G具有足够大的最小次数的假设替换G具有足够大的色数的假设会发生什么。这与排除一起$K_t$，不足以保证两个反完备子图都具有较大的最小度；但它有效，如果而不是排除$K_t$我们排除完整的二部图$K_{t,t}$。更准确地说：对于所有人$t,C\ge 1$那里存在$d\ge 1$使得如果G的最小度至少为d ，并且不包含完整的二部图$K_{t,t}$作为一个子图，则有两个反完全子图，其最小度至少为c。