Article

A Coloring Algorithm for Triangle-Free Graphs

01/2011;
Source: arXiv

ABSTRACT We give a randomized algorithm that properly colors the vertices of a
triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors,
where \Delta(G) is the maximum degree of G. The algorithm takes
O(n\Delta2(G)log\Delta(G)) time and succeeds with high probability, provided
\Delta(G) is greater than log^{1+{\epsilon}}n for a positive constant
{\epsilon}. The number of colors is best possible up to a constant factor for
triangle-free graphs. As a result this gives an algorithmic proof for a sharp
upper bound of the chromatic number of a triangle-free graph, the existence of
which was previously established by Kim and Johansson respectively.

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• Source
Article:Coloring Graphs with Sparse Neighborhoods
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ABSTRACT: It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d2/f is at most O(d/log f). This is tight (up to a constant factor) for all admissible values of d and f.
Journal of Combinatorial Theory, Series B.
• Article:On an upper bound of a graph's chromatic number, depending on the graph's degree and density.
J. Comb. Theory, Ser. B. 01/1977; 23:247-250.
• Chapter:On Colouring the Nodes of a Network
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ABSTRACT: Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node. Suppose also that no connected component of N is an n-simplex. Then it is possible to colour the nodes of N with n colours so that no two nodes of the same colour are joined.
06/2010: pages 118-121;

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Keywords

algorithm

algorithmic proof

chromatic number

constant factor

Johansson

maximum degree

n vertices

positive constant

randomized algorithm

triangle-free graph

triangle-free graph G

triangle-free graphs

vertices