A new sufficient condition for the existence of 3-kernels
Let D be a digraph and k be a positive integer. We say a subset N of V(D) is a k-kernel of D if it is both k-independent and (k − 1)-absorbent. A short chord of a closed trail C = (v0, v1, . . . , vt) is an arc a = (vi, vj) which does not belong to C and the distance from vi to vj in C is exactly two. The spacing between two chords e = (u, v) and f = (x, y) in C is the distance from u to x in C. A set of chords in a closed trail C has an odd spacing if at least two chords have an odd spacing. In this work, we prove that if D is a strongly connected digraph where every odd cycle has a short chord and every even closed trail has three short chords with an odd spacing, then D has a 3-kernel.
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