Realizing degree sequences as Z3-connected graphs

An integer-valued sequence π = (d1, . . . , dn) is graphic if there is a simple graph G with degree sequence of π . We say the π has a realization G. Let Z3 be a cyclic group of order three. A graph G is Z3-connected if for every mapping b : V (G) → Z3 such that  v∈V (G) b(v) = 0, there is an orientation of G and amapping f : E(G) → Z3−{0} such that for each vertex v ∈ V (G), the sum of the values of f on all the edges leaving from v minus the sum of the values of f on the all edges coming to v is equal to b(v). If an integer-valued sequenceπ has a realization Gwhich is Z3-connected, thenπ has a Z3-connected realization G. Let π = (d1, . . . , dn) be a nonincreasing graphic sequence with dn ≥ 3.We prove in this paper that if d1 ≥ n − 3, then π has a Z3-connected realization unless the sequence is (n − 3, 3n−1) or is (k, 3k) or (k2, 3k−1) where k = n − 1 and n is even; if dn−5 ≥ 4, then π has a Z3-connected realization unless the sequence is (52, 34) or (5, 35). © 2014 Elsevier B.V. All rights reserved.