A sequence S is potentially KmββPkβ graphical if it has a realization
containing a KmββPkβ as a subgraph. Let Ο(KmββPkβ,n) denote
the smallest degree sum such that every n-term graphical sequence S with
Ο(S)β₯Ο(KmββPkβ,n) is potentially KmββPkβ graphical.
In this paper, we prove that Ο(KmββPkβ,n)β₯(2mβ6)nβ(mβ3)(mβ2)+2, for nβ₯mβ₯k+1β₯4. We conjecture that
equality holds for nβ₯mβ₯k+1β₯4. We prove that this conjecture is
true for m=k+1=5 and m=k+2=5.Comment: 5 page
A sequence S is potentially Kp,1,1β graphical if it has a realization
containing a Kp,1,1β as a subgraph, where Kp,1,1β is a complete
3-partite graph with partition sizes p,1,1. Let Ο(Kp,1,1β,n) denote
the smallest degree sum such that every n-term graphical sequence S with
Ο(S)β₯Ο(Kp,1,1β,n) is potentially Kp,1,1β graphical. In
this paper, we prove that Ο(Kp,1,1β,n)β₯2[((p+1)(nβ1)+2)/2] for
nβ₯p+2. We conjecture that equality holds for nβ₯2p+4. We prove
that this conjecture is true for p=3.Comment: 5 page