Geodesic
ℱ-WORM colorings of some 2-trees: partition vectors
Ars Mathematica Contemporanea, Tome 16 (2019) no. 1, pp. 173-182.

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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.
DOI : 10.26493/1855-3974.1017.0ac
Keywords: 2-tree, maximal outerplanar, partition, Stirling numbers 
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Julian D. Allagan; Vitaly Voloshin. ℱ-WORM colorings of some 2-trees: partition vectors. Ars Mathematica Contemporanea, Tome 16 (2019) no. 1, pp. 173-182. doi : 10.26493/1855-3974.1017.0ac. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1017.0ac/

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