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Let D=(V(D),A(D)) be a digraph. A subset S?V(D) is k-independent if the distance between every pair of vertices of S is at least k, and it is ?-absorbent if for every vertex u in V(D)?S there exists vS such that the distance from u to v is less than or equal to ?. A k-kernel is a k-independent and (k?1)-absorbent set. A kernel is simply a 2-kernel.A classical result due to Duchet states that if every directed cycle in a digraph D has at least one symmetric arc, then D has a kernel. We propose a conjecture generalizing this result for k-kernels and prove it true for k=3 and k=4.  相似文献   
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A (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.  相似文献   
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DP-coloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph G with a list-assignment L to finding an independent transversal in an auxiliary graph with vertex set ◂{}▸{(v,c)|◂+▸vV(G),◂+▸cL(v)}. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks’ type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.  相似文献   
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A digraph D is supereulerian if D has a spanning closed ditrail. Bang‐Jensen and Thomassé conjectured that if the arc‐strong connectivity of a digraph D is not less than the independence number , then D is supereulerian. A digraph is bipartite if its underlying graph is bipartite. Let be the size of a maximum matching of D. We prove that if D is a bipartite digraph satisfying , then D is supereulerian. Consequently, every bipartite digraph D satisfying is supereulerian. The bound of our main result is best possible.  相似文献   
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