On r-Hued Coloring of Hypercubes

超立方体的r-Hued染色

For positive integers k and r,a(k,r)-coloring of graph G is a proper vertex k-coloring of G such that the neighbors of any vertex v∈V(G)receive at least min{d_(G)(v),r}different colors.The r-hued chromatic number of G,denoted χ_(r)(G),is the smallest integer k such that G admits a(k,r)-coloring.Let Q_(n) be the n-dimensional hypercube.For any integers n and r with n≥2 and 2≤r≤5,we investigated the behavior of χ_(r)(Q_(n)),and determined the exact value of χ_(2)(Q_(n))and χ_(3)(Q_(n))for all positive integers n.

对于正整数k和r,图G的一个(k,r)-染色是指图G中顶点的一个正常k-染色,并且每个顶点v的领域有至少d_(G)(v)或r种不同颜色.图G的r-hued染色数是最小的整数k使得G有一个(k,r)-染色,记作χ_(r)(G).令Q_(n)为n维超立方体.对于任意整数n和r,其中n≥2,2≤r≤5,研究了χ_(r)(Q_(n)),并确定了χ_(2)(Q_(n))和χ_(3)(Q_(n))对于所有正整数n的精确值.