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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; The Inside View</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="page4.xml">Objectives</a>
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        <a class="toc_link" href="">Foundations</a>
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        <a class="toc_link" href="page5.xml">What is a Graph?</a>
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        <a class="toc_link" href="page7.xml">Edges and Vertices</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="page12.xml">Types of Graphs</a>
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        <a class="toc_link" href="page21.xml">Trees</a>
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        <a class="toc_link" href="">Solving the Drilling Problem</a>
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        <a class="toc_link" href="page24.xml">Independent Sets, Cliques, etc.</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="page38.xml">What's Next?</a>
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    <div id="content"><h1>The Inside View</h1>
				<h2>The &quot;Inside View&quot;</h2>
				<h2>How far did you get?</h2>
				<p>
					You probably found this approach to be quite difficult.
				</p>
				<p>
					Let's try the problem from the other &quot;end&quot;, and see if we
					have more success.
				</p>
				<p>
					Let's look at the points in the
					<b>middle</b> of the figure.  It is clear
					that these points are &quot;close&quot; to one another, and therefore
					form a clique.  The size of this clique is a lower bound for
					the chromatic number.
				</p>
				<p>
					Let us try to find as large a clique as possible.
				</p>
				<div class="box">
					
						<div class="figure" align="center"><a name=""/><img src="../graphics/fig12.gif" alt="The Largest Clique"/><br/><b><i>Figure 28:
		The Largest Clique</i></b><br/><i>
								The 24 vertices inside the yellow-shaded area comprise
								a clique.  This is the largest clique of the graph.
								<b>Corollary:</b> Any drilling tool that
								satisfies the given constraints must use at least 24
								passes.
							</i><br/><br/></div>
					
				</div>
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					<p>
					Thus, 24 is our new lower bound!
					</p>
					
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