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- W4299013936 abstract "A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $binom{[k]}{2}$ 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 $tau_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $pge Cn^{-1/4}ln^{9/4}n$, $tau_2(G_{n,p})=(2+o(1))chi(G_{n,p})$ where $chi$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $tau_2(G_{n,p}) = lceil{{sqrt{8Delta+1} +5}/{2}}rceil$ where $Delta$ represents the maximum degree. For the more general concept of $t$-tone coloring, we achieve similar results." @default.
- W4299013936 created "2022-10-02" @default.
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- W4299013936 date "2012-10-01" @default.
- W4299013936 modified "2023-10-11" @default.
- W4299013936 title "The t-tone chromatic number of random graphs" @default.
- W4299013936 doi "https://doi.org/10.48550/arxiv.1210.0635" @default.
- W4299013936 hasPublicationYear "2012" @default.
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