For a simple connected graph G = (V (G),E(G)) and a positive integer k, a radio k-labelling of G is a mapping \(f \colon V(G)\rightarrow \{0,1,2,\ldots \}\) such that \(|f(u)-f(v)|\geqslant k+1-d(u,v)\) for each pair of distinct vertices u and v of G, where d(u,v) is the distance between u and v in G. The radio k-chromatic number is the minimum span of a radio k-labeling of G. In this article, we study the radio k-labelling problem for complete m-ary trees T_m,h and determine the exact value of radio k-chromatic number for these trees when k ≥ 2h − 1.

中文翻译：

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全三叉树的射电 k 色数

对于简单连通图G = ( V ( G ), E ( G )) 和正整数k，G的单选k标记是映射\(f \colon V(G)\rightarrow \{0,1 ，如图2所示，\ ldots \} \） ，使得\（| F（u）的-f（v）的| \ geqslant第k + 1 d（U，v）\）对于每对独特的顶点ü和v的ģ，其中ð（Û，v）之间的距离为ù和v在ģ。所述无线电K-色数是G的单选k标记的最小跨度。在本文中，我们研究了无线电ķ完整-labelling问题米进制树Ť_米，ħ和确定的无线电精确值ķ对于这些树-chromatic号码时ķ ≥2 ħ - 1。