tailieunhanh - Báo cáo toán học: "Total 4-choosability of series-parallel graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Total 4-choosability of series-parallel graphs. | Total 4-choosability of series-parallel graphs Douglas R. Woodall School of Mathematical Sciences University of Nottingham Nottingham NG7 2RD UK Submitted Jan 25 2005 Accepted Oct 18 2006 Published Oct 31 2006 Mathematics Subject Classification 05C15 Abstract It is proved that if G is a K4-minor-free graph with maximum degree 3 then G is totally 4-choosable that is if every element vertex or edge of G is assigned a list of 4 colours then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results this shows that the List-Total-Colouring Conjecture that ch00 G x G for every graph G is true for all K4-minor-free graphs and therefore for all outerplanar graphs. Keywords Outerplanar graph Minor-free graph Series-parallel graph List total colouring. 1 Introduction We use standard terminology as defined in the references for example 8 or 10 . We distinguish graphs which are always simple from multigraphs which may have multiple edges however our theorem is only for graphs. For a graph or multigraph G its edge chromatic number total vertex-edge chromatic number edge choosability or list edge chromatic number total choosability and maximum degree are denoted by x0 G X00 G ch0 G ch00 G and A G respectively. So ch00 G is the smallest k for which G is totally k-choosable. There is great interest in discovering classes of graphs G for which the choosability or list chromatic number ch G is equal to the chromatic number x G . The List-EdgeColouring Conjecture LECC and List-Total-Colouring Conjecture LTCC 1 4 6 are that for every multigraph H ch0 H x0 H and ch00 H x00 H respectively so the conjectures are that ch G x G whenever G is the line graph or the total graph of a multigraph H. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R97 1 For an outerplanar simple graph H Wang and Lih 9 proved that ch0 H x H A H if A