Author:
Albertson Michael O.,Berman David M.
Abstract
The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the followingTheorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].
Publisher
Cambridge University Press (CUP)
Reference8 articles.
1. Map Color Theorem
2. On colouring the nodes of a network
3. Every planar graph has an acyclic 7-coloring
4. Every planar graph has an acyclic 8-coloring
5. A proof of B. Grünbaum's conjecture on the acyclic 5-colorability of planar graphs (Russian);Borodin;Dokl. Akad. Nauk SSSR,1976
Cited by
8 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Hardness transitions and uniqueness of acyclic colouring;Discrete Applied Mathematics;2024-03
2. Graphs on Higher Surfaces;Graph Coloring Problems;2011-10-28
3. Adjacency posets of planar graphs;Discrete Mathematics;2010-03
4. Layout of Graphs with Bounded Tree-Width;SIAM Journal on Computing;2005-01
5. The Game Coloring Number of Planar Graphs;Journal of Combinatorial Theory, Series B;1999-03