Packing Chromatic Number of Base-3 Sierpiński Graphs

On S-packing edge-colorings of cubic graphs

“…The vertex analogous of S-packing edge-coloring has been first studied by Goddard and Xu [16,17] and then recently on cubic graphs [2,3,4,12,14]. The particular case of (1, 2, .…”

Packing Chromatic Number of Subdivisions of Cubic Graphs

“…By the argument above, in particular, by (5), colors 3 a and 3 b are not used on vertices in B = {u 1 , u 2 , u 3 , u 4 , u 5 , u 7 , u 8 , u 9 , u 11 , u 13 }. If at least one of them, say 3 a , is also not used on A, then after recoloring u with 3 a , we obtain a coloring satisfying (c).…”

“…The concept has attracted a considerable attention recently: there are around 30 papers on the topic (see e.g. [1,3,4,5,6,7,8,9,10,11,12,13,22] and references in them). In particular, Fiala and Golovach [10] proved that finding the packing chromatic number of a graph is NP-hard even in the class of trees.…”

Packing coloring of Sierpiński-type graphs

“…is equivalent to the inequality 2 n+1 > (4 − k) · k n , we get a n < a n+1 for any k ≥ 4. This means that the sequence (a n ) n∈N is strictly increasing as soon as k ≥ 4 (naturally, for k = 3 this is not the case, as the result of [5] also shows). Thus (χ ρ (S n k )) n∈N is unbounded when k > 3.…”

“…As the situation of the packing chromatic number of the standard Sierpiński graphs is in some sense resolved by Theorem 1 and the results of [5], it is interesting to consider natural generalizations of this class of graphs. One such class are the generalized Sierpiński graphs, whose definition we now recall.…”