LetV fin andE fin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesV inf andE inf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveE inf ⫋V inf ⫋V fin ⫋E fin. Finally we consider labellings by positive integers and characterize the class corresponding toV inf.
AMS subject classification (1980)05 C 55 05 C 38
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