Abstract
Given non-negative integers \(r, s,\) and \(t,\) an \([r,s,t]\)-coloring of a graph \(G = (V(G),E(G))\) is a mapping \(c\) from \(V(G) \cup E(G)\) to the color set \(\{1,\ldots ,k\}\) such that \(\left|c(v_i) - c(v_j)\right| \ge r\) for every two adjacent vertices \(v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s\) for every two adjacent edges \(e_i,e_j,\) and \(\left|c(v_i) - c(e_j)\right| \ge t\) for all pairs of incident vertices and edges, respectively. The \([r,s,t]\)-chromatic number \(\chi _{r,s,t}(G)\) of \(G\) is defined to be the minimum \(k\) such that \(G\) admits an \([r,s,t]\)-coloring. In this note we examine \(\chi _{1,1,t}(K_p)\) for complete graphs \(K_p.\) We prove, among others, that \(\chi _{1,1,t}(K_p)\) is equal to \(p+t-2+\min \{p,t\}\) whenever \(t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,\) but is strictly larger if \(p\) is even and sufficiently large with respect to \(t.\) Moreover, as \(p \rightarrow \infty \) and \(t=t(p),\) we asymptotically have \(\chi _{1,1,t}(K_p)=p+o(p)\) if and only if \(t=o(p).\)
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This research was supported in part by the Hungarian Scientific Research Fund, OTKA Grant 81493.
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Kemnitz, A., Marangio, M. & Tuza, Z. \([1,1,t]\)-Colorings of Complete Graphs. Graphs and Combinatorics 29, 1041–1050 (2013). https://doi.org/10.1007/s00373-012-1153-3
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DOI: https://doi.org/10.1007/s00373-012-1153-3