ℱ-WORM colorings of some 2-trees: partition vectors

Abstract

Suppose ℱ = {F₁, …, F_t} is a collection of distinct subgraphs of a graph G = (V, E). An ℱ-WORM coloring of G is the coloring of its vertices such that no copy of each subgraph F_i ∈ ℱ is monochrome or rainbow. This generalizes the notion of F-WORM coloring that was introduced recently by W. Goddard, K. Wash, and H. Xu. A (restricted) partition vector (ζ_α, …, ζ_β) is a sequence whose terms ζ_r are the number of ℱ-WORM colorings using exactly r colors, with α ≤ r ≤ β. The partition vectors of complete graphs and those of some 2-trees are discussed. We show that, although 2-trees admit the same partition vector in classic proper vertex colorings which forbid monochrome K₂, their partition vectors in K₃-WORM colorings are different.