Denote this relation by formula_149. For example, let "formula_137", then "formula_1" can reduce to "formula_152" since "formula_138" is free for "formula_1". A word "formula_19" is said to be locked if "formula_19" has no free letter; hence "formula_19" can not be reduced. Given patterns "formula_158", if "formula_1" reduces to "formula_83" and "formula_83" reduces to "formula_162", then "formula_1" reduces to "formula_162". Denote this relation by formula_165. A pattern "formula_1" is unavoidable if and only if "formula_1" reduces to a word of length one; hence "formula_168" such that "formula_169" and "formula_170". Given a simple graph "formula_171", a edge coloring "formula_172" matches pattern "formula_1" if there exists a simple path "formula_174" in "formula_175" such that the sequence "formula_176" matches "formula_1". Otherwise, "formula_178" is said to avoid "formula_1" or be "formula_1"-free. Similarly, a vertex coloring "formula_181" matches pattern "formula_1" if there exists a simple path "formula_183" in "formula_175" such that the sequence "formula_185" matches "formula_1". The pattern chromatic number formula_187 is the minimal number of distinct colors needed for a "formula_1"-free vertex coloring "formula_178" over the graph "formula_175". Let "formula_191" where "formula_192" is the set of all simple graphs with a maximum degree no more than "formula_193". Similarly, formula_194 and formula_195 are defined for edge colorings. A pattern "formula_1" is avoidable on graphs if "formula_197" is bounded by "formula_198", where "formula_198" only depends on "formula_1". There exists an absolute constant formula_178, such that "formula_208" for all patterns "formula_1" with "formula_210".