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# The t-Tone Chromatic Number of Random Graphs

journal contribution

posted on 2013-03-06, 00:00 authored by Deepak Bal, Patrick Bennett, Andrzej Dudek, Alan FriezeAlan FriezeA proper 2-tone *k*-coloring of a graph is a labeling of the vertices with elements from ([k]2) such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph *G*, denoted *τ* 2(*G*) is the smallest *k* such that*G* admits a proper 2-tone *k* coloring. In this paper, we prove that w.h.p. for p≥Cn−1/4ln9/4n,τ2(Gn,p)=(2+o(1))χ(Gn,p) where χ represents the ordinary chromatic number. For sparse random graphs with *p* = *c*/*n*, *c* constant, we prove that τ2(Gn,p)=⌈(8Δ+1−−−−−−√+5)/2 where Δ represents the maximum degree. For the more general concept of *t*-tone coloring, we achieve similar results.