H-Kernels by Walks

Let D = ( V , E ) and H = ( U , F ) be digraphs and consider a colouring of the arcs of D with the vertices of H ; we say that D is H coloured. We study a natural generalisation of the notion of kernel, as introduced by V. Neumann and Morgenstern (1944), to prove that If every cycle of D is an H -cycle, then D has an H -kernel by walks . As a consequence of this, we are able to give several sufficient conditions for the existence of H -kernels by walks; in particular, we solve partially a conjecture by Bai et al. in this context [ 2 ]; viz., they work with complete H without loops, and use paths rather than walks, so whenever the existence of H -paths is implied by the existence of H -walks our result can be use to corroborate Bai’s conjecture—in particular, if D is two coloured, and each cycle is alternating, then each alternating walk contains an alternating path.