A characterization of 4-$χ_S$-vertex-critical graphs for $S\succeq (1,3,3,3,\ldots)$

Abstract

If S = (s 1 , s 2 , . . .) is a non-decreasing sequence of positive integers, then the Spacking k-coloring of a graph G is a mapping c :In this paper, 4-χ S -vertex-critical graphs are characterized for sequences S = (1, s 2 , s 3 , . . .) with s 2 ≥ 3. There are 28 sporadic examples and two infinite families of such graphs.