Brook's type upper bounds for coloring parameters of infinite graphs and Konig's Lemma

We prove that Brook's type theorems for several coloring parameters of infinite graphs, which are true in ZFC, are not provable in ZF (i.e., the Zermelo-Fraenkel set theory without the Axiom of Choice (AC)). In ZF, we apply Konig's Lemma (a weak form of AC) and give a combinatorial argument to find a general upper bound of the total distinguishing number for any connected infinite graph with finite maximum degree. In ZF, we give new combinatorial arguments to formulate new conditions for the existence of distinguishing chromatic number, distinguishing chromatic index, total chromatic number, odd chromatic number, and neighbor-distinguishing index in infinite locally finite connected graphs, which are equivalent to Konig's Lemma (any infinite locally finite connected graph has a ray).