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			Home &gt; Graph Theory: Industrial Drilling
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					&gt; Trees</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="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>Trees</h1>
				<div class="figure" align="center"><a name="img_fig_tree_b"/><img src="../graphics/fig_tree_b.gif"/><br/><b><i>Figure 22</i></b><br/><br/></div>
				<p>
				If
				<m:math>
					<m:mi>G</m:mi>
					<m:mfenced>
						<m:mi>V</m:mi>
						<m:mi>E</m:mi>
					</m:mfenced>
				</m:math> is a 3-regular graph, then
				<m:math>
					<m:mo>|</m:mo>
						<m:mi>V</m:mi>
					<m:mo>|</m:mo>
					<m:mo>≥</m:mo>
					<m:mn>4</m:mn>
				</m:math>. 
				Figure <a href="#22">22</a> shows two spanning
				trees without common edges (their edges are shown,
				respectively, as red and blue) in a complete graph with 4
				vertices.  Suppose
				<m:math>
					<m:mi>G</m:mi>
					<m:mfenced>
						<m:mi>V</m:mi>
						<m:mi>E</m:mi>
					</m:mfenced>
				</m:math> is a graph with
				<m:math>
					<m:mo>|</m:mo>
						<m:mi>V</m:mi>
					<m:mo>|</m:mo>
					<m:mo>&gt;</m:mo>
					<m:mn>4</m:mn>
				</m:math> and all vertices of degree 3. 
				</p>
				
					<p><div class="problem"><a name="problem20"/><b>Problem
				20:
			</b><br/>
						
						<p>
						Can <m:math><m:mi>G</m:mi></m:math>, as described above, have two spanning trees without common edges?
						</p>
						
					</div></p>
					<p><div class="problem"><a name="problem21"/><b>Problem
				21:
			</b><br/>
						
							<p>
								Does your conclusion hold for the
								<i>3-regular</i> graph on 4 vertices?
							</p>
						
						<a target="_new" href="answer16.xml">Answer</a>
					</div></p>
				
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