tailieunhanh - Báo cáo toán học: "Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition. | Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition Xueliang Li and Fengxia Liu Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071 . China lxl@ xjulfx@ Submitted Jan 2 2008 Accepted Oct 5 2008 Published Oct 20 2008 Mathematics Subject Classifications 05C05 05C15 05C70 05C35 Abstract The monochromatic tree partition number of an r-edge-colored graph G denoted by tr G is the minimum integer k such that whenever the edges of G are colored with r colors the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general to determine this number is very difficult. For 2-edge-colored complete multipartite graph Kaneko Kano and Suzuki gave the exact value of t2 K rn n2 nk . In this paper we prove that if n 3 and K n n is 3-edge-colored such that every vertex has color degree 3 then 13 K n n 3. Keywords monochromatic tree tree partition number complete bipartite graph 3-edge-colored color degree 1 Introduction The monochromatic tree partition number or simply tree partition number of an r-edge-colored graph G denoted by tr G which was introduced by Erdos Gyarfas and Pyber 1 is the minimum integer k such that whenever the edges of G are colored with r colors the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. Erdos Gyarfas and Pyber 1 conjectured that the tree partition number of an r-edge-colored complete graph is r 1. Moreover they proved that the conjecture is true for r 3. For the case r 2 it is equivalent to the fact that for any graph G either G or its complement is connected an old remark of Erdos and Rado. Supported by NSFC PCSIRT and the 973 program. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R131 1 For infinite complete graph Hajnal 2 proved that the tree partition number for an r-edge-colored infinite complete graph is at most r. For finite complete graph Haxell and Kohayakawa 3 proved that any .