"A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma(K_4-e,n)$ denote the smallest degree sum such that every $n$ -term graphical sequence $S$ with $\sigma(S)\geq\sigma(K_4-e,n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$ . In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$ for $n\geq7$ and $n=4,5$ , and $\sigma(K_4-e,6)=20$ .''

"In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$ -term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$ . Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$ -term graphical sequence $S$ having $\sigma(S)\geq l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma(H,n)$ . We show $\sigma(C_{2m+1},n)=m(2n-m-1)+2$ , for $m\geq 3$ , $n\geq 3m$ ; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$ , for $m\geq 3$ , $n\geq 5m-2$ .''

In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.