Acyclic coloring of graphs

Alon, Noga; McDiarmid, Colin; Reed, Bruce

doi:10.1002/rsa.3240020303

Cited by 208 publications

(215 citation statements)

References 5 publications

(3 reference statements)

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“…Concerning graphs having a fixed maximum degree, most of the results on this topic come from Alon et al [AMR90], where the following results were proved: (1) asymptotically, there exist graphs of maximum degree ∆ with acyclic chromatic number in Ω( ∆ 4 3 (log ∆) 1 3 ) ; (2) asymptotically, it is possible to acyclically color any graph of maximum degree ∆ with O(∆ 4…”

Section: Introductionmentioning

confidence: 99%

Acyclic coloring of graphs of maximum degree five: Nine colors are enough

Fertin

Raspaud

2008

Information Processing Letters

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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. In this paper, we show that any graph of maximum degree 5 has acyclic chromatic number at most 9, and we give a linear time algorithm that achieves this bound.

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Section: Introductionmentioning

confidence: 99%

Acyclic coloring of graphs of maximum degree five: Nine colors are enough

Fertin

Raspaud

2008

Information Processing Letters

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“…In Section 2, we have shown that for any fixed digraph G, the queuenumber of every L k (G) is at most the tree-queuenumber of G. Also the underlying graph of L k (G) has bounded star chromatic number, since the maximum outdegree and indegree of L k (G) are equal to those of G, respectively, and it has been shown that graphs with bounded maximum degree have bounded star chromatic number [1,10]. Thus, we have the following theorem.…”

Section: Three-dimensional Drawings Of Iterated Line Digraphsmentioning

confidence: 99%

Laying Out Iterated Line Digraphs Using Queues

Hasunuma

2004

Graph Drawing

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Abstract. In this paper, we study a layout problem of a digraph using queues. The queuenumber of a digraph is the minimum number of queues required for a queue layout of the digraph. We present upper and lower bounds on the queuenumber of an iterated line digraph L k (G) of a digraph G. In particular, our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the result on the queuenumber of L k (G), it is shown that for any fixed digraph G, L k (G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L k (G). We also apply these results to particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, butterfly digraphs, and wrapped butterfly digraphs.

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“…Albertson and Berman mentioned in [1] that Erdős proved that a(r) = Ω(r 4/3− ) and conjectured that a(r) = o(r 2 ). Alon, McDiarmid and Reed [5] sharpened Erdős' lower bound to a(r) ≥ c r 4/3 /(log r) 1/3 and proved that a(r) ≤ 50 r 4/3 .…”

Section: Introductionmentioning

confidence: 98%