Precoloring extension for 2-connected graphs with maximum degree three

Abstract

Let G = G ( V , E ) be a simple graph, L a list assignment with | L ( v ) | = Δ ( G ) ∀ v ∈ V and W ⊆ V an independent subset of the vertex set. Define d ( W ) ≔ min { d ( v , w ) ∣ v , w ∈ W } to be the minimum distance between two vertices of W . In this paper it is shown that if G is 2-connected with Δ ( G ) = 3 and G is not K 4 then every precoloring of W is extendable to a proper list coloring of G provided that d ( W ) ≥ 6 . An example shows that the bound is sharp. This result completes the investigation of precoloring extensions for graphs with | L ( v ) | = Δ ( G ) for all v ∈ V where the precolored set W is an independent set.