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Acrylic improper colorings of graphs
Author(s) -
Boiron P.,
Sopena E.,
Vignal L.
Publication year - 1999
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(199909)32:1<97::aid-jgt9>3.0.co;2-o
Subject(s) - combinatorics , mathematics , colored , edge coloring , monochromatic color , planar graph , vertex (graph theory) , greedy coloring , discrete mathematics , graph , graph power , line graph , materials science , botany , composite material , biology
In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex‐coloring in which adjacent vertices are allowed to have the same color, but each color class V i satisfies some condition depending on i . Such a coloring is acyclic if there are no alternating 2‐colored cycles. We prove that every outerplanar graph can be acyclically 2‐colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999