2-Distance choosability of planar graphs with a restriction for maximum degree

Abstract

A proper k-coloring of a graph G is a 2-distance k-coloring of G if each pair of vertices with distance no more than 2 are colored differently. We call G is 2-distance L-colorable if it has a 2-distance coloring π such that π(v)∈L(v), where L={L(v)∣v∈V} is a list assignment of G. Similarly, G is called to be 2-distance k-choosable if there is a 2-distance L-coloring of G such that any list assignment L satisfies |L(v)|≥k for each v∈V(G). The 2-distance list chromatic number of G, denoted by χ2l(G), is the minimum positive integer k such that G is 2-distance k-choosable. In this paper, we prove that every planar graph G with maximum degree Δ has χ2l(G)≤18 if Δ≤5, and χ2l(G)≤4Δ−3 if Δ≥6.