## Acyclic Colourings of 1-Planar Graphs (1999)

Citations: | 5 - 0 self |

### BibTeX

@MISC{Borodin99acycliccolourings,

author = {O.V. Borodin and A. V. Kostochka and A. Raspaud and E. Sopena},

title = {Acyclic Colourings of 1-Planar Graphs},

year = {1999}

}

### OpenURL

### Abstract

. A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20. Keywords. Acyclic colouring, planar graphs, 1-planar graphs. 1

### Citations

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(Show Context)
Citation Context ... of a graph is acyclic if every cycle uses at least three colors (Grunbaum [9]). The acyclic chromatic number of G, denoted by a(G), is the minimum k such that G admits an acyclic k-coloring. Borodin =-=[5]-=- proved Grunbaum's conjecture that every planar graph is acyclically 5colorable. This bound is best possible. Moreover, there are bipartite 2-degenerate planar graphs G with a(G) = 5 (Kostochka and Me... |

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Citation Context ...o, Hakimi, Mitchem and Schmeichel [10, p.38-39] proved that E(G) can be partitioned into a(G) star forests (whose every component is a star). Using [5], this confirms the conjecture of Algor and Alon =-=[1]-=- that the edges of every planar graph can be partitioned into five star forests. In this paper we study the acyclic coloring of 1-planar graphs. Our main result is the following: Theorem 1 Every 1-pla... |

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Citation Context ...pes of colorings. Suppose one has proved that a(G)sa. Then G has the star chromatic number at most a2 a\Gamma1 (Grunbaum [9]) and the oriented chromatic number at most a2 a\Gamma1 (Raspaud and Sopena =-=[13]-=-); every m-coloring of the edges of G can be homomorphically mapped on that of a graph with at most am a\Gamma1 vertices (Alon and Marshall [2]); every m-coloring of the edges of mixed graph G combine... |

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Citation Context ... difference is that Euler's formula for it says jV j \Gamma jEj + jF js1. Accordingly, the extensions of Theorem 1 and Corrolaries 2-6 to the projective plane also take place. Alon, Mohar and Sanders =-=[3]-=- showed that the acyclic 5-colorability of the plane graphs easily implies the acyclic 7-colorability of the projective plane graphs. 2 Proof of Theorem 7 Let P 0 be a counterexample with the fewest v... |

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Citation Context ...ssing diagonals into each 4-face. Every G vf arises similarly from a plane quadrangulation. In 1965, Ringel [14, 15] conjectured that each 1-planar graph is 6-colorable; this was confirmed by Borodin =-=[6]-=- in 1984. (A new simplier proof was given in [7].) It follows, the vertices and faces of each plane graph can be colored with 6 colors so that every two adjacent or incident elements have different co... |

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Citation Context ...nals added cannot be colored acyclically with fewer than 7 colors. Theorem 1 has a number of applications to other coloring problems, listed below. For the precise definitions of the notions used see =-=[2, 9, 10, 12, 13]-=-. Corollary 2 Every plane graph has an acyclic simultaneous coloring of vertices and faces with at most 20 colors. Corollary 3 Every 1-planar graph has star chromatic number at most 20 \Delta 2 19 . C... |

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Citation Context ...hromatic number at most a2 a\Gamma1 (Raspaud and Sopena [13]); every m-coloring of the edges of G can be homomorphically mapped on that of a graph with at most am a\Gamma1 vertices (Alon and Marshall =-=[2]-=-); every m-coloring of the edges of mixed graph G combined with its n-coloring can be homomorphically mapped on a graph with at most a(2n +m) a\Gamma1 vertices (Nesetril and Raspaud [12]). Also, Hakim... |

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Citation Context ...runbaum's conjecture that every planar graph is acyclically 5colorable. This bound is best possible. Moreover, there are bipartite 2-degenerate planar graphs G with a(G) = 5 (Kostochka and Mel'nikov, =-=[11]-=-). However, if a plane graph G has no cycles of length less than 5, then a(G)s4, and if it has no cycles of length less than 7, then a(G)s3 (Borodin, Kostochka and Woodall [8]). Acyclic colorings turn... |

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Citation Context ...n and Marshall [2]); every m-coloring of the edges of mixed graph G combined with its n-coloring can be homomorphically mapped on a graph with at most a(2n +m) a\Gamma1 vertices (Nesetril and Raspaud =-=[12]-=-). Also, Hakimi, Mitchem and Schmeichel [10, p.38-39] proved that E(G) can be partitioned into a(G) star forests (whose every component is a star). Using [5], this confirms the conjecture of Algor and... |

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Citation Context ...ever x and y are adjacent in G. A 1-planar graph is a graph which can be drawn on the plane so that every edge crosses at most one other edge. One of the reasons for introducing this notion by Ringel =-=[14]-=- is that the graph G vf of adjacency/incidence of the vertices and faces of each plane graph G is 1-planar. (On a map, take a capital of each country and join adjacent capitals by a railroad via a com... |

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Citation Context ...adjacent or incident elements have different colors. The 3-prism needs 6 colors for such a coloring, because it has 11 elements to be colored, and no three of them can be colored the same. Archdeacon =-=[4]-=- proved that the vertices and faces of each bipartite plane graph are simultaneously 5-colorable. A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors (Grunbaum [9]... |

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Citation Context ...ses similarly from a plane quadrangulation. In 1965, Ringel [14, 15] conjectured that each 1-planar graph is 6-colorable; this was confirmed by Borodin [6] in 1984. (A new simplier proof was given in =-=[7]-=-.) It follows, the vertices and faces of each plane graph can be colored with 6 colors so that every two adjacent or incident elements have different colors. The 3-prism needs 6 colors for such a colo... |

1 |
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Citation Context ...ostochka and Mel'nikov, [11]). However, if a plane graph G has no cycles of length less than 5, then a(G)s4, and if it has no cycles of length less than 7, then a(G)s3 (Borodin, Kostochka and Woodall =-=[8]-=-). Acyclic colorings turned out to be useful for obtaining results about other types of colorings. Suppose one has proved that a(G)sa. Then G has the star chromatic number at most a2 a\Gamma1 (Grunbau... |

1 |
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Citation Context ...[4] proved that the vertices and faces of each bipartite plane graph are simultaneously 5-colorable. A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors (Grunbaum =-=[9]-=-). The acyclic chromatic number of G, denoted by a(G), is the minimum k such that G admits an acyclic k-coloring. Borodin [5] proved Grunbaum's conjecture that every planar graph is acyclically 5color... |

1 |
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Citation Context ...aborative Research Grant n o 97-1519. 1 graph whose faces have size 3 or 4 by adding two crossing diagonals into each 4-face. Every G vf arises similarly from a plane quadrangulation. In 1965, Ringel =-=[14, 15]-=- conjectured that each 1-planar graph is 6-colorable; this was confirmed by Borodin [6] in 1984. (A new simplier proof was given in [7].) It follows, the vertices and faces of each plane graph can be ... |