A characterization for a graphic sequence to have a realization containing a desired cycle - Yin, Jian-Hua; Wang, Ye (Utilitas Mathematica, 2011)
Let r ≥ 3 and S(r) be the set of all circular arrangements of 1,2,⋯ ,r. It is well known that S(r) = i1i 2⋯⋯.⋯..ir|i1 = 1 and i2i3⋯⋯⋯⋯.ir is an arrangement of 2,3,⋯ ,r and |S(r)| = (r -1)(r 2)-l. Let α = i 1i2⋯⋯⋯⋯ir ⋯S(r) and φ = (d1d2⋯dn) be a graphic sequence with n ≥ r. If ir has a realization G with vertex set V(G) = 1,2, ⋯,n such that dG(i) = di for i = 1,2, ⋯,n and i1i2⋯..iri1 is a cycle of length r in G, then φ is said to be potentially C rα-graphic. We use A(α) to denote the set of all potentially Crα-graphic sequences. In this paper, we give a characterization for φ ⋯∩α⋯S(τ) A(α). In other words, we characterize φ such that φ is potentially Grα-graphic for each α⋯ S(r).