D(3)-vertex-distinguishable proper edge-chromatic number of C_m·P_n
Tian Jing-jing · Journal of Lanzhou University of Technology · 2009
Let G(V,E) be a connected graph with order not less than three,α,β be positive integers,f be a mapping from E(G) on to {1,2,…,α}.If f were an α-proper edge-coloring such that C(u)≠C(v) whenever u,v∈V(G) with 1≤d(u,v)≤β,then f would be called α-D(β)-vertex-distinguishable proper edge-coloring of graph G(shortly,an α-D(β)-VDPEC of G) and the necessary minimum number χ′β-vd(G)=min{α|G has an αD(β)-VDPEC} would be called D(β)-vertex-distinguishable edge-chromatic number of G,in which d(u,v) denotes the minimum distance between two points. The D(3)-vertex-distinguishable proper edge-coloring chromatic number of Cm·Pn could be obtained thereby.