Adjacent vertex distinguishing edge choosability of 1-planar graphs with maximum degree at least 23

Abstract

For a simple graph G, an adjacent vertex distinguishing edge k-coloring of G is a mapping ϕ: E(G)→{1,2,…,k} such that ϕ(e1)≠ϕ(e2) for every adjacent edges e1 and e2 in E(G) and Cϕ(u)≠Cϕ(v) for each edge uv∈E(G), where Cϕ(v) (or Cϕ(u)) denotes the set of colors assigned to the edges incident with v (or u). For each edge e∈E(G), let L(e) be a list of possible colors that can be used on e. If, whenever we give a list assignment L={L(e)||L(e)|=k,e∈E(G)}, there exists an adjacent vertex distinguishing edge k-coloring ϕ such that ϕ(e)∈L(e) for each edge e∈E(G), then we say that ϕ is a list adjacent vertex distinguishing edge k-coloring. The smallest k for which such a coloring exists is called the adjacent vertex distinguishing edge choosability of G, denoted by cha′(G). A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. There is almost no result yet about cha′(G) if G is a 1-planar graph. We prove that cha′(G)≤Δ+3 for every 1-planar graph G with maximum degree Δ≥23.