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

Julian D. Allagan, Vitaly Voloshin

Abstract


Suppose ℱ = {F1, …, Ft} 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 Fi ∈ 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 (ζα, …, ζβ) is a sequence whose terms ζr 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.


Keywords


2-tree, maximal outerplanar, partition, Stirling numbers

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DOI: https://doi.org/10.26493/1855-3974.1017.0ac

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications