On relationship between Hamiltonian path and holes in L(3,2,1)-coloring of minimum span

The L(2,1)-coloring and L(3,2,1)-coloring on a finite simple connected graph G originate from frequency assignment problem. Any L(2,1)-coloring on G of minimum possible span is a λ2,1-coloring of G and its span is the λ2,1-number of G, denoted by λ2,1(G). Any integer i, 0