Abstract A (k, r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d, r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k, r)-coloring. Let f ( r ) = r + 3 if 1 ≤ r ≤ 2, f ( r ) = r + 5 if 3 ≤ r ≤ 7 and f ( r ) = ⌊ 3 r / 2 ⌋ + 1 if r ≥ 8. In [Discrete Math., 315-316 (2014) 47-52], an extended conjecture of Wegner is proposed that if G is planar, then χr(G) ≤ f(r); and this conjecture was verified for K4-minor free graphs. For an integer n ≥ 4, let K4(n) be the set of all subdivisions of K4 on n vertices. We obtain decompositions of K4(n)-minor free graphs with n ∈ {5, 6, 7}. The decompositions are applied to show that if G is a K4(7)-minor free graph, then χr(G) ≤ f(r) if and only if G is not isomorphic to K6.

中文翻译：

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K4(7)-minor free graphs的分解和r-hued着色

摘要 图 G 的 (k, r) 着色是 G 的适当 k 顶点着色，使得度数为 d 的每个顶点的邻居将接收至少 min{d, r} 种不同的颜色。用 χr(G) 表示的 r 色调色数是图 G 具有 (k, r) 着色的最小整数 k。设 f ( r ) = r + 3 如果 1 ≤ r ≤ 2, f ( r ) = r + 5 如果 3 ≤ r ≤ 7 并且 f ( r ) = ⌊ 3 r / 2 ⌋ + 1 如果 r ≥ 8. In [Discrete Math., 315-316 (2014) 47-52]，提出了Wegner的扩展猜想，如果G是平面的，则χr(G) ≤ f(r)；并且这个猜想在 K4-minor 自由图上得到了验证。对于整数 n ≥ 4，令 K4(n) 是 K4 在 n 个顶点上的所有细分的集合。我们获得 K4(n)-minor free graphs 的分解，其中 n ∈ {5, 6, 7}。应用分解表明，如果 G 是 K4(7)-次要自由图，