2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 389396
author:  Guillaume Fertin and André Raspaud 

title: 
Acyclic Coloring of Graphs of Maximum Degree
Δ

keywords:  Acyclic chromatic number, acyclic coloring algorithm, maximum degree 
abstract: 
An acyclic coloring of a graph
G
is a coloring of its vertices such that: (i) no two
neighbors in
G
are assigned the same color and (ii) no bicolored
cycle can exist in
G
. The acyclic chromatic number of
G
is the least number of colors necessary to
acyclically color
G
, and is denoted by
a(G)
. We show that any graph of maximum degree
Δ
has acyclic chromatic number at most
Δ(Δ1) / 2
for any
Δ≥5
, and we give an
O(nΔ
algorithm to acyclically color any graph of maximum
degree
2
)
Δ
with the above mentioned number of colors. This
result is roughly two times better than the best general
upper bound known so far, yielding
a(G)≤Δ(Δ1) +2
[albert]. By a deeper study of the case
Δ=5
, we also show that any graph of maximum degree 5 can
be acyclically colored with at most 9 colors, and give a
linear time algorithm to achieve this bound.

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reference: 
Guillaume Fertin and André Raspaud (2005), Acyclic
Coloring of Graphs of Maximum Degree
Δ
, in 2005 European Conference on Combinatorics,
Graph Theory and Applications (EuroComb '05), Stefan
Felsner (ed.), Discrete Mathematics and Theoretical
Computer Science Proceedings AE, pp. 389396

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