The Singular Chromatic Number of a Graph

A proper coloring of a graph G is an assignment of colors to the vertices of G such that adjacent vertices are assigned distinct colors. The minimum number of colors in a proper coloring of G is the chromatic number chi(G) of G. For a graph G and a proper coloring c : V(G) -> {1, 2, ..., k} of the vertices of G for some positive integer k, the color code of a vertex v of G (with respect to c) is the ordered pair code(v) = (c(v), S-v), where S-v = {c(u) : u is an element of N(v)}. The coloring c is singular if distinct vertices have distinct color codes and the singular chromatic number chi(si),(G) of G is the minimum positive integer k for which G has a singular k-coloring. Thus chi(G) <= chi(si)(G) <= n for every graph G of order n. A characterization is established for all triples (a, b, n) of positive integers for which there exists a graph G of order n with chi(G) = a and chi(si)(G) = b. It is shown for every vertex v and every edge e in a graph G that chi(si)(G) - 1 <= chi(si)(G - v) <= chi(si)(G) + deg v and chi(sj)(G) - 1 <= chi(si)(G - e) <= chi(si)(G) + 2 and that all these four bounds are sharp. We also determine the singular chromatic numbers of cycles and paths.