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			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; Modeling it as a Graph Theory Problem</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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    <div id="content"><h1>Modeling it as a Graph Theory Problem</h1>
				<h2>Modeling The Drilling Problem As A Graph Theory Problem</h2>
				<h2>Review</h2>
				<p>
				To be sure you are ready to tackle this graph theory problem, 
				take a moment to review the basic definitions.
				</p>
				<p>
				Make sure, in particular, that you fully understand the definitions 
				of an independent set, a clique, and of vertex coloring.
				</p>
				<p>
				To check your understanding, prove the following statements:
				</p>
				
					<p><div class="problem"><a name="problem26"/><b>Problem
				26:
			</b><br/>
						
							<p>
								A graph
								<m:math>
									<m:mi>G</m:mi>
									<m:mfenced>
										<m:mi>V</m:mi>
										<m:mi>E</m:mi>
									</m:mfenced>
								</m:math>
								is k-chromatic if there is a partitioning of
								<m:math>
									<m:mi>V</m:mi>
								</m:math> into
								<m:math>
									<m:mi>k</m:mi>
								</m:math>
								subsets
								<m:math>
									<m:msub>
										<m:mi>V</m:mi>
										<m:mn>1</m:mn>
									</m:msub>
									<m:mo>,</m:mo>
									<m:msub>
										<m:mi>V</m:mi>
										<m:mn>2</m:mn>
									</m:msub>
									<m:mo>,</m:mo>
									<m:mo>…</m:mo>
									<m:mo>,</m:mo>
									<m:msub>
										<m:mi>V</m:mi>
										<m:mi>k</m:mi>
									</m:msub>
								</m:math> such that every 
								<m:math>
									<m:msub>
										<m:mi>V</m:mi>
										<m:mi>i</m:mi>
									</m:msub></m:math>
								is an independent set of
								<m:math>
									<m:mi>G</m:mi>
								</m:math>.
							</p>
						
					</div></p>
					<p><div class="problem"><a name="problem27"/><b>Problem
				27:
			</b><br/>
						
							<p>
								If
								<m:math>
									<m:mi>ω</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
								</m:math>
								is the size of the maximum clique in a graph
								<m:math>
									<m:mi>G</m:mi>
								</m:math> and
								<m:math>
									<m:mi>χ</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
								</m:math>
								is the chromatic number of
								<m:math>
									<m:mi>G</m:mi>
								</m:math>, then
							</p>
							<p><div class="equation" align="center">
								<m:math>
									<m:mi>χ</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
									<m:mo>≥</m:mo>
									<m:mi>ω</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
								</m:math>
							<br/><b><i>Equation
				12</i></b></div></p>
						
					</div></p>
					<p><div class="problem"><a name="problem28"/><b>Problem
				28:
			</b><br/>
						
							<p>
								If
								<m:math>
									<m:mi>α</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
								</m:math>
								is the size of the maximum independent set in a graph
								<m:math>
									<m:mi>G</m:mi>
								</m:math>
								with
								<m:math>
									<m:mi>n</m:mi>
								</m:math>
								vertices and chromatic number
								<m:math>
									<m:mi>χ</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
								</m:math>, then
							</p>
							<p><div class="equation" align="center">
								<m:math>
									<m:mi>χ</m:mi>
									<m:mfenced>
										<m:mi>G</m:mi>
									</m:mfenced>
									<m:mo>≥</m:mo>
									<m:mfrac>
										<m:mi>n</m:mi>
										<m:mrow>
											<m:mi>α</m:mi>
											<m:mfenced>
												<m:mi>G</m:mi>
											</m:mfenced>
										</m:mrow>
									</m:mfrac>
								</m:math>
							<br/><b><i>Equation
				13</i></b></div></p>
						
					</div></p>
				
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