Realizing degree sequences as Z3Z3-connected graphs

An integer-valued sequence π=(d1,…,dn)π=(d1,…,dn) is graphic   if there is a simple graph GG with degree sequence of ππ. We say the ππ has a realization GG. Let Z3Z3 be a cyclic group of order three. A graph GG is Z3Z3-connected   if for every mapping b:V(G)→Z3b:V(G)→Z3 such that ∑v∈V(G)b(v)=0∑v∈V(G)b(v)=0, there is an orientation of GG and a mapping f:E(G)→Z3−{0}f:E(G)→Z3−{0} such that for each vertex v∈V(G)v∈V(G), the sum of the values of ff on all the edges leaving from vv minus the sum of the values of ff on the all edges coming to vv is equal to b(v)b(v). If an integer-valued sequence ππ has a realization GG which is Z3Z3-connected, then ππ has a Z3Z3-connected realization  GG. Let π=(d1,…,dn)π=(d1,…,dn) be a nonincreasing graphic sequence with dn≥3dn≥3. We prove in this paper that if d1≥n−3d1≥n−3, then ππ has a Z3Z3-connected realization unless the sequence is (n−3,3n−1)(n−3,3n−1) or is (k,3k)(k,3k) or (k2,3k−1)(k2,3k−1) where k=n−1k=n−1 and nn is even; if dn−5≥4dn−5≥4, then ππ has a Z3Z3-connected realization unless the sequence is (52,34)(52,34) or (5,35)(5,35).