We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if G is such a graph, then G 2 is (Δ( G ) + 72)-degenerate. This implies an upper bound of Δ( G ) ∣ 73 on the chromatic number of G 2 as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon-Tarsi number, and correspondence chromatic number. We also show that if Δ( G ) is sufficiently large, then the upper bounds on each of these parameters of G 2 can all be lowered to Δ( G ) + 2 (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let S be a finite list of positive integers, with 4 ∉ S . For each constant C , we construct a planar graph G s,c with no cycle with length in S , but for which χ (G S,C 2 ) >Δ( G s,c ) + C .

中文翻译：

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无四环平面图平方的简并与着色

我们证明了对没有 4 循环的平面图的正方形着色的几个结果。首先，我们证明如果 G 是这样的图，则 G 2 是 (Δ( G ) + 72)-退化的。这意味着 G 2 的色数以及色数的几种变体（例如列表色数、油漆数、Alon-Tarsi 数和对应色数）的上限 Δ( G ) ∣ 73。我们还表明，如果 Δ( G ) 足够大，那么 G 2 的这些参数中的每一个的上限都可以降低到 Δ( G ) + 2（这是最好的）。为了补充这些结果，我们表明 4-cycles 在具有此属性方面是独一无二的。具体来说，让 S 是一个有限的正整数列表，其中 4 ∉ S 。对于每个常数 C ，我们构造一个平面图 G s,c ，在 S 中没有长度的循环，但对于其中 χ (GS,C 2 ) >Δ( G s,c ) + C 。