TY - JOUR
T1 - Acyclic coloring of graphs
AU - Alon, Noga
AU - Mcdiarmid, Colin
AU - Reed, Bruce
PY - 1991
Y1 - 1991
N2 - A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two‐colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdös who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two‐colored cycle. All the proofs rely heavily on probabilistic arguments.
AB - A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two‐colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdös who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two‐colored cycle. All the proofs rely heavily on probabilistic arguments.
UR - http://www.scopus.com/inward/record.url?scp=84990717256&partnerID=8YFLogxK
U2 - 10.1002/rsa.3240020303
DO - 10.1002/rsa.3240020303
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AN - SCOPUS:84990717256
VL - 2
SP - 277
EP - 288
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
SN - 1042-9832
IS - 3
ER -