List graphs and distance-consistent node labelings

In this paper we consider node labelings c of an undirected connected graph G=(V,E) with labels {1,2,... ,|V|}, which induce a list distance c(u,v)=|c(v)-c(u)| besides the usual graph distance d(u,v). Our main aim is to find a labeling c so c(u,v) is as close to d(u,v) as possible. For any graph we specify algorithms to find a distance-consistent labeling, which is a labeling c that minimize $\sum\limits_{u,v\in V} (c(u,v)-d(u,v)) ^2$. Such labeliings may provide structure for very large graphs. Furthermore, we define a labeling c fulfilling $d(u_1,v_1)labeling, and a graph that has a list labeling is a list graph. We prove that list graphs exist for all n=|V| and all k=|E| :n-1\leq k\leq n(n-1)/2, and establish basic properties. List graphs are hamiltonian, and show weak versions of properties of path graphs.