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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; The Best 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="page10.xml">A First Theorem</a>
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        <a class="toc_link" href="">Solving 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="page31.xml">First Lower Bound</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>The Best Coloring</h1>
				<h2>25 is the right answer!</h2>
				<div class="figure" align="center"><a name=""/><img src="../graphics/fig15.gif"/><br/><b><i>Figure 34</i></b><br/><i>
					Consider the 31 points in the middle of the figure on the right.
					<m:math><m:mn>19</m:mn></m:math> of them are shown as yellow diamonds; the remainder are shown as red stars.
					</i><br/><br/></div>
				<div class="box">
					<b><i>Proposition 1:</i></b>
					
					<p>
					The distance between any two diamonds is less than 35.
					</p>
					
				</div>
				<div class="box">
					<b><i>Proposition 2:</i></b>
					
					<p>
					The distance between any diamond and any star is less than 35.
					</p>
					
				</div>
				<h2>Corollaries:</h2>
				<ol><li>
					For any coloring of <m:math><m:mi>G</m:mi></m:math>, the 19 diamonds use 19 distinct colors.
					</li><li>
					For any coloring of <m:math><m:mi>G</m:mi></m:math>, the 19 diamonds use colors different from those used for the
					<m:math><m:mn>12</m:mn></m:math> stars.
					</li><li>
					If <m:math><m:mi>h</m:mi></m:math> is the number of colors needed to color the 12 stars, 
					the number of colors needed for coloring <m:math><m:mi>G</m:mi></m:math> is at least
					</li></ol>
				<p><div class="equation" align="center">
					<m:math>
						<m:mn>19</m:mn>
						<m:mo>+</m:mo>
						<m:mi>h</m:mi>
					</m:math>
				<br/><b><i>Equation
				15</i></b></div></p>
				<div class="figure" align="center"><a name=""/><img src="../graphics/fig16.gif"/><br/><b><i>Figure 35</i></b><br/><i>
						The three green stars are the only points at a distance
						greater than 35 from the red star. Hence, only these
						points can be colored the same color as the red star.  The
						distance between any two of the green stars is less than
						35, however.  Thus any coloring of the twelve diamonds
						uses any color at most twice, implying that
						<m:math>
							<m:mi>h</m:mi>
							<m:mo>≥</m:mo>
							<m:mfrac>
								<m:mn>12</m:mn>
								<m:mn>2</m:mn>
							</m:mfrac>
							<m:mo>=</m:mo>
							<m:mn>6</m:mn>
						</m:math>
					</i><br/><br/></div>
				<div class="box">
					
					<p>
					Therefore, the chromatic number of <m:math><m:mi>G</m:mi></m:math> is 25.
					</p>
					
				</div>
				<p>
				We now know that the coloring just given is indeed a best coloring.
				</p>
				<p>
				Using this scheme, we can drill the workpiece in only 25 passes.
      			Furthermore, we need only 3 different configurations of the gang drill's 
      			bits, corresponding to each of the groups of color classes.
      			</p>
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