A t-frugal colouring of a graph G is an assignment of colours to the vertices of G, such that each colour appears at most t times in the neighbourhood of any vertex. A dichotomy theorem for the complexity of deciding whether a graph has a 1-frugal colouring with k colours was found by McCormick and Thomas, and then later extended to restricted graph classes by Kratochvil and Siggers. We generalize the McCormick and Thomas theorem by proving a dichotomy theorem for the complexity of deciding whether a graph has a t-frugal colouring with k colours, for all pairs of positive integers t and k. We also generalize bounds of Lih et al. for the number of colours needed in a 1-frugal colouring of a given \(K_4\)-minor-free graph with maximum degree \(\Delta \) to t-frugal colourings, for any positive integer t.

甲吨-节俭着色图的ģ是颜色的顶点的分配ģ，使得每个颜色至多出现吨倍任一顶点的附近。McCormick 和 Thomas 发现了一个二分定理，用于决定一个图是否具有k种颜色的 1-frugal 着色，然后由 Kratochvil 和 Siggers 扩展到受限图类。我们通过证明二分定理来概括麦考密克和托马斯定理，该定理用于决定一个图是否具有k种颜色的t -frugal 着色，对于所有正整数对t和k. 我们还概括了 Lih 等人的界限。对于任何正整数t，对于具有最大度数\(\Delta \)的给定\(K_4\) -minor-free 图的 1-frugal 着色所需的颜色数为t -frugal 着色。