2 research outputs found
List graphs and distance-consistent node labelings
In this paper we consider node labelings c of an undirected connected graph Gβ=β(V,βE) with labels {1,β2,β...,ββ£Vβ£}, which induce a list distance c(u,βv)β=ββ£c(v)β
ββ
c(u)β£ besides the usual graph distance d(u,βv). Our main aim is to find a labeling c so c(u,βv) is as close to d(u,βv) as possible. For any graph we specify algorithms to find a distance-consistent labeling, which is a labeling c that minimize . Such labeliings may provide structure for very large graphs. Furthermore, we define a labeling c fulfilling d(u1,βv1)β<βd(u2,βv2)βββc(u1,βv1)ββ€βc(u2,βv2) for all node pairs u1,βv1 and u2,βv2 as a list labeling, and a graph that has a list labeling is a list graph. We prove that list graphs exist for all nβ=ββ£Vβ£ and all kβ=ββ£Eβ£β:βnβ
ββ
1ββ€βkββ€βn(nβ
ββ
1)/2, and establish basic properties. List graphs are hamiltonian, and show weak versions of properties of path graphs.</p