Improper sum-list colouring of 2-trees

Abstract

Let G be a graph and P be a class of graphs. We consider list colouring of vertices of G in which the sizes of lists assigned to different vertices can be different. We colour G from the lists in such a way that vertices of each colour class induce a graph in P. The smallest possible sum of all the list sizes, such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes is denoted by χscP(G).Let D1, I1 be classes of acyclic and triangle-free graphs, respectively. Given a graph G, we consider a graph HG whose vertex set is the set of triangles of G and whose two vertices t1,t2 are adjacent if the triangles t1, t2 share an edge in G.In this work we provide an upper bound of n+α(HG) on χscD1(G), where G is an arbitrary n-vertex 2-tree and α(HG) is the independence number of HG. We give a polynomial-time algorithm to find the assignment of list sizes that realizes this bound. Moreover, we prove that this bound is achieved for an arbitrary n-vertex 2-tree G that is 3-sun-free (equivalently, Hajoś-graph-free, see Fig. 7) and for such an n-vertex 2-tree G whose graph HG is a tree with α(HG)=⌈n2⌉−1. Finally, we show that χscI1(G)=χscD1(G) for each 2-tree G.