Some undecidable problems involving the edge-coloring and vertex-coloring of graphs

Abstract Certain problems involving the coloring the edges or vertices of infinite graphs are shown to be undecidable. In particular, G and H be finite 3-connected graphs, or triangles. Then a doubly-periodic infinite graph F is constructed such that the following problem is undecidable: For a coloring of a finite subset of the edges of F red and blue, determine whether this 2-coloring can be extended to all the edges of F without either a red G or blue H occurring. In the case of vertex-coloring, a similar result holds; here, three colors are used, and the forbidden configuration is (as usual) simply two adjacent vertices of the same color.