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Báo cáo toán học: "A note on graph coloring extensions and list-colorings"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A note on graph coloring extensions and list-colorings | A note on graph coloring extensions and list-colorings Maria Axenovich Department of Mathematics Iowa State University Ames IA 50011 USA axenovic@math.iastate.edu Submitted Oct 24 2002 Accepted Feb 10 2003 Published Mar 23 2003 MR Subject Classifications 05C15 Abstract Let G be a graph with maximum degree A 3 not equal to K All and let P be a subset of vertices with pairwise distance d P between them at least 8. Let each vertex x be assigned a list of colors of size A if x E V P and 1 if x E P. We prove that it is possible to color V G such that adjacent vertices receive different colors and each vertex has a color from its list. We show that d P cannot be improved. This generalization of Brooks theorem answers the following question of Albertson positively If G and P are objects described above can any coloring of P in at most A colors be extended to a proper coloring of G in at most A colors We say that a vertex-coloring of a graph G V E is proper if the colors used on adjacent vertices are distinct. For an assignment of a color set typically called a list l x to each vertex x E V we say that vertices are colored from their lists by a coloring c if c x E l x for each x E V c is called a list-coloring of G. A coloring c of V G extends a coloring c of vertices in P if it is a proper coloring with c x d x for each x E P. We denote by dG x the degree of x in a graph G and by G X the subgraph of G induced by a set of vertices X. The classic Brooks theorem states that any simple connected graph G with maximum degree A can be colored properly in at most A colors unless G Ka- or G is an odd cycle. Recently Albertson posed the following question. Take a graph described above precolor a fixed set of vertices P in A colors arbitrarily. Under what condition on P can we extend that coloring to a proper coloring of G in at most A colors He asks whether this condition is a large distance between the vertices in P . Albertson noticed though that the maximum degree of a graph .