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    <div id="titlebar">Graph Theory: Industrial Drilling</div>
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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; A Good Coloring</div>
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        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
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        <a class="toc_link" href="page26.xml">Vertex Coloring</a>
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        <a class="toc_link" href="page28.xml">Modeling it as a Graph Theory Problem</a>
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        <a class="toc_link" href="page31.xml">First Lower Bound</a>
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        <a class="toc_link" href="page33.xml">The Inside View</a>
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        <a class="toc_link" href="page34.xml">A Good Coloring</a>
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        <a class="toc_link" href="page37.xml">The Best Coloring</a>
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    <div id="content"><h1>A Good Coloring</h1>
				<h2>The solution below uses 25 colors.</h2>
				<p>
					... which is only one above our just-established lower bound of
					<m:math>
						<m:mn>24</m:mn>
					</m:math>.
				</p>
				<p>
				Our coloring uses three groups of <i>color classes</i>.
				Each of these groups of color classes corresponds to a
				different configuration of drill bits on the gang drill.
				</p>
				<div class="figure" align="center"><a name=""/><img src="../graphics/plane7.gif"/><br/><b><i>Figure 29</i></b><br/><i>
					The first group consists of just one color class.
					You will recognize if as the independent set with 7 vertices; 
					the largest independent set in the graph.
					</i><br/><br/></div>
				<div class="figure" align="center"><a name=""/><img src="../graphics/fig13.gif"/><br/><b><i>Figure 30</i></b><br/><i>
					The second group has 12 color classes, each of size 4.
					A class from this group - four yellow vertices - is shown on the right.
					</i><br/><br/></div>
				<div class="figure" align="center"><a name=""/><img src="../graphics/finalyellow.gif"/><br/><b><i>Figure 31</i></b><br/><i>
					Every other class in the second group is obtained from this class by moving it either right 
					and/or down.
					The yellow vertices in the figure on the left comprise the union of these color classes.
					</i><br/><br/></div>
				<div class="figure" align="center"><a name=""/><img src="../graphics/fig14.gif"/><br/><b><i>Figure 32</i></b><br/><i>
					The third group of color classes is represented by an independent set comprised 
					of 3 green vertices in the figure to the right.
					There are 12 classes in the third group, each of size 3.
					</i><br/><br/></div>
				<div class="figure" align="center"><a name=""/><img src="../graphics/finalpattern.gif"/><br/><b><i>Figure 33</i></b><br/><i>
					Similarly to the second group, every class from the third group is obtained from the 
					green class by moving it down and/or to the right.
					</i><br/><br/></div>
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