tailieunhanh - Báo cáo toán học: "Color Neighborhood Union Conditions for Long Heterochromatic Paths in Edge-Colored Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Color Neighborhood Union Conditions for Long Heterochromatic Paths in Edge-Colored Graphs. | Color Neighborhood Union Conditions for Long Heterochromatic Paths in Edge-Colored Graphs He Chen and Xueliang Li Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071 China lxl@ Submitted Apr 12 2007 Accepted Nov 1 2007 Published Nov 12 2007 Mathematics Subject Classifications 05C38 05C15 Abstract Let G be an edge-colored graph. A heterochromatic rainbow or multicolored path of G is such a path in which no two edges have the same color. Let CN v denote the color neighborhood of a vertex v of G. In a previous paper we showed that if CN u u CN v s color neighborhood union condition for every pair of vertices u and v of G then G has a heterochromatic path of length at least b2 5 4J. In the present paper we prove that G has a heterochromatic path of length at least dẠp e and give examples to show that the lower bound is best possible in some sense. Keywords edge-colored graph color neighborhood heterochromatic rainbow or multicolored path. 1. Introduction We use Bondy and Murty 3 for terminology and notations not defined here and consider simple graphs only. Let G V E be a graph. By an edge-coloring of G we will mean a function C E N the set of natural numbers. If G is assigned such a coloring then we say that G is an edge-colored graph. Denote the edge-colored graph by G C and call C e the color of the edge e 2 E. We say that C uv 0 if uv 2 E G for U V 2 V G . For a subgraph H of G we denote C H C e I e 2 E H g and c H C H . For a vertex v of G the color neighborhood CN v of v is defined as the set C e I e is incident with vg and the color degree is dc v CN v . A path is called heterochromatic rainbow or Research supported by NSFC PCSIRT and the 973 program. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R77 1 multicolored if any two edges of it have different colors. If u and v are two vertices on a path P uPv denotes the segment of P from u to v whereas vP 1u denotes the same segment but from v to u. There are many existing .