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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; First Lower Bound</div>
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        <a class="toc_link" href="page31.xml">First Lower Bound</a>
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    <div id="content"><h1>First Lower Bound</h1>
				<h2>Finding A Lower Bound</h2>
				<h2>A First Attempt</h2>
				<p>
				At this point, we can already establish some kind of lower bound for the number of colors needed:
				</p>
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					<p>
					Every coloring of the graph G must use at least 13 colors.
					</p>
					
				</div>
				<h2>Proof</h2>
				<p>
				Since the size of every independent set is at most 7, the number of colors used by 
				any coloring <b>must</b> be at least
				</p>
				<p><div class="equation" align="center">
					<m:math>
						<m:mfrac>
							<m:mn>91</m:mn>
							<m:mn>7</m:mn>
						</m:mfrac>
						<m:mo>=</m:mo>
						<m:mn>13</m:mn>
					</m:math>
				<br/><b><i>Equation
				14</i></b></div></p>
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