Compatible powers of Hamilton cycles in dense graphs

The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph G = (V , E ) , an incompatibility system .F over G is a family .F = { F (v) }( v epsilon V )such that for every v is an element of V , F-v is a family of edge pairs {e , e ' } is an element of ( E (G) (2) ) with e boolean AND e ' = { v } . Moreover, for an integer k is an element of N , we say .F is k - bounded if for every vertex v and its incident edge e , there are at most k pairs in F-v containing e . Krivelevich, Lee and Sudakov proved that there is an universal constant,mu. > 0 such that for every Dirac graph G and every ,. n - bounded incompatibility system .F over G , there exists a Hamilton cycle C superset of G where every pair of adjacent edges e , e ' of C satisfies {e, e ' }(soc) F-v for {v} = e boolean AND e ' . This resolves a conjecture posed by H & auml;ggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to .F ). We study high powers of Hamilton cycles in this context and show that for every y > 0 and k is an element of N , there exists a constant ,mu > 0 such that for sufficiently large n is an element of N and every ,. n - bounded incompatibility system over an n - vertex graph G with delta (G) >= ( k/k +1 +gamma) n , there exists a compatible k th power of a Hamilton cycle in G . Moreover, we give a mu n - bounded construction which k has minimum degree k/k+1 n + Omega(n) and contains nocompatible k th power of a Hamilton cycle.