  A sequence $S$ is potentially $K_{m}-P_{k}$ graphical if it has a realization
containing a $K_{m}-P_{k}$ as a subgraph. Let $\sigma(K_{m}-P_{k}, n)$ denote
the smallest degree sum such that every $n$-term graphical sequence $S$ with
$\sigma(S)\geq \sigma(K_{m}-P_{k}, n)$ is potentially $K_{m}-P_{k}$ graphical.
In this paper, we prove that $\sigma (K_{m}-P_{k}, n)\geq
(2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq k+1\geq 4.$ We conjecture that
equality holds for $n \geq m \geq k+1\geq 4.$ We prove that this conjecture is
true for $m=k+1=5$ and $m=k+2=5$.
