F-WORM colorings of some 2-trees: partition vectors - Allagan, Julian D. (Ars mathematica contemporanea, 2019)
Suppose ?$\mathscr{F} = \{F_1, …, F_t\}$? is a collection of distinct subgraphs of a graph ?$G = (V, E)$?. An ?$\mathscr{F}$?-WORM coloring of ?$G$? is the coloring of its vertices such that no copy of each subgraph ?$F_i \in \mathscr{F}$? is monochrome or rainbow. This generalizes the notion of ?$F$?-WORM coloring that was introduced recently by W. Goddard et al. Congr. Numerantium 219, 161--173 (2014). A (restricted) partition vector ?$(ζ_{α}, …, ζ_{β})$? is a sequence whose terms ?$ζ_r$? are the number of ?$\mathscr{F}$?-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 ?$K_2$?, their partition vectors in ?$K_3$?-WORM colorings are different.
Keywords: 2-drevo; 2-tree; maksimalen zunanjeravninski; maximal outerplanar; partition; razdelitev; Stirling numbers; Stirlingova števila