A kernel in a directed graph D = (V, A) is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V \ K there is a vertex u in K, such that (v, u) is an arc of D. It is well known that the problem of the existence of a kernel is NP-complete for a general digraph. Bang-Jensen and Gutin pose an interesting problem (Problem 12.3.5) in their book [Digraphs: Theory, Algorithms and Applications, London: Springer-Verlag, 2000]: to characterize all circular digraphs with kernels. In this paper, we study the problem of the existence of the kernel for several special classes of circular digraphs. Moreover, a class of counterexamples is given for the Duchet kernel conjecture (for every connected kernel-less digraph which is not an odd directed cycle, there exists an arc which can be removed and the obtained digraph is still kernel-less).

This paper is committed to dealing with the measure of noncompactness of operators in Banach spaces. First, we give a characterization of the measure of noncompact Hausdorff operators with respect to the Hausdorff metric. Then, we show a formula of the Hausdorff measure of noncompactness of operators in . Finally, several common equivalent measures of noncompactness of operators and related proofs are provided.

A cycle of length 4 is called a quadrilateral and a multigraph is called standard if every edge in it has multiplicity at most 2. A quadrilateral with four multiedges is called heavy-quadrilateral. It is proved that if the minimum degree of M is at least 6k-2, then M contains k vertex-disjoint quadrilaterals, such that k-1 of them are heavy-quadrilaterals and the remaining one is a quadrilateral with three multiedges, with only three exceptions.

In this paper, we prove an existence and uniqueness theorem of the solution for strongly pseudomonotone variational inequalities in reflexive Banach spaces. Based on this result, and investigate the stability behavior of the perturbed variational inequalities. Moreover, we obtain an existence theorem of solutions for strongly quasimonotone variational inequalities in finite dimensional spaces.