S-packing chromatic critical paths and cycles
Abstract
Let S=(s1,s2,…) be a non-decreasing sequence of positive integers. For a graph G with vertex set V(G), a labeling φ V(G) \1,…,k\ is an S-packing k-coloring if, whenever two distinct vertices u,v∈ V(G) are assigned the same color i, their distance in G is greater than si. The minimum k for which G admits such a coloring is the S-packing chromatic number of G. A graph G is S-vertex-critical if S(G-v) < S(G) for every v ∈ V(G), and it is S-critical if S(H) < S(G) holds for every proper subgraph H of G. In this paper, the exact value of S(Pn) is determined for every path of order n and for every packing sequence S where si < 2i holds for each entry si. As a consequence, S-critical and S-vertex-critical paths are identified for each such sequence S. In addition, we extend earlier results on S-critical cycles and provide a complete characterization of S-critical and S-vertex-critical cycles for packing sequences S= (1, s2, … ) with s2 ∈ \2,3\ and s3,s4 ∈ \4,5,6,7\.
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