Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.
Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.
In this paper, a new trust region algorithm for nonlinear equality constrained LC1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subprobiem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.
Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.
