Overfull graph

<h2 id="examples">Examples</h2>
<p>Every <a href="/facts/Parity_(mathematics)/7zq2UM4V">odd</a> <a href="/facts/Cycle_graph/bYhnPDHc">cycle graph</a> of length three or more is overfull. The product of its degree (two) and half its length (rounded down) is one less than the number of edges in the cycle. More generally, every <a href="/facts/Regular_graph/j9dSoLE6">regular graph</a> with an odd number 
  
    
      
        n
      
    
    {\displaystyle n}
  
 of vertices is overfull, because its number of edges, 
  
    
      
        Δ
        n
        
          /
        
        2
      
    
    {\displaystyle \Delta n/2}
  
 (where 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
 is its degree), is larger than 
  
    
      
        Δ
        ⌊
        n
        
          /
        
        2
        ⌋
      
    
    {\displaystyle \Delta \lfloor n/2\rfloor }
  
.
</p>
<h2 id="properties">Properties</h2>
<p>A few properties of overfull graphs:
</p>
<ol><li>Overfull graphs are of odd order.</li>
<li>Overfull graphs are <a href="/facts/Edge_coloring/AcdpUxCv">class 2</a>. That is, they require at least Δ + 1 colors in any <a href="/facts/Edge_coloring/AcdpUxCv">edge coloring</a>.</li>
<li>A graph <i>G</i>, with an overfull subgraph <i>S</i> such that 
  
    
      
        
          Δ
          (
          G
          )
          =
          Δ
          (
          S
          )
        
      
    
    {\displaystyle \displaystyle \Delta (G)=\Delta (S)}
  
, is of class 2.</li></ol>
<h2 id="overfull-conjecture">Overfull conjecture</h2>
<p>In 1986, <a href="/facts/Amanda_Chetwynd/hJYk0Ug6">Amanda Chetwynd</a> and <a href="/facts/Anthony_Hilton/z8AJ5216">Anthony Hilton</a> posited the following <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> that is now known as the overfull conjecture.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
A graph <i>G</i> with 
  
    
      
        Δ
        (
        G
        )
        >
        n
        
          /
        
        3
      
    
    {\displaystyle \Delta (G)>n/3}
  
 is class 2 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it has an overfull subgraph S such that 
  
    
      
        
          Δ
          (
          G
          )
          =
          Δ
          (
          S
          )
        
      
    
    {\displaystyle \displaystyle \Delta (G)=\Delta (S)}
  
.
<p>This conjecture, if true, would have numerous implications in graph theory, including the <a href="/facts/1-factorization_conjecture/2hx3n7c3">1-factorization conjecture</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="algorithms">Algorithms</h2>
<p>For graphs in which 
  
    
      
        Δ
        ≥
        n
        
          /
        
        3
      
    
    {\displaystyle \Delta \geq n/3}
  
, there are at most three <a href="/facts/Induced_subgraph/G3EUC0MQ">induced</a> overfull subgraphs, and it is possible to find an overfull subgraph in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a>. When 
  
    
      
        Δ
        ≥
        n
        
          /
        
        2
      
    
    {\displaystyle \Delta \geq n/2}
  
, there is at most one induced overfull subgraph, and it is possible to find it in <a href="/facts/Linear_time/77T62gmf">linear time</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Chetwynd, A. G.; Hilton, A. J. W. (1986), "Star multigraphs with three vertices of maximum degree" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 100 (2): 303–317, doi:10.1017/S030500410006610X, MR 0848854. <a href="https://eprints.lancs.ac.uk/id/eprint/19968/1/download2.pdf" target="_blank">https://eprints.lancs.ac.uk/id/eprint/19968/1/download2.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Chetwynd, A. G.; Hilton, A. J. W. (1989), "1-factorizing regular graphs of high degree—an improved bound", Discrete Mathematics, 75 (1–3): 103–112, doi:10.1016/0012-365X(89)90082-4, MR 1001390. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Niessen, Thomas (2001), "How to find overfull subgraphs in graphs with large maximum degree. II", Electronic Journal of Combinatorics, 8 (1), Research Paper 7, MR 1814514. <a href="http://www.combinatorics.org/Volume_8/Abstracts/v8i1r7.html" target="_blank">http://www.combinatorics.org/Volume_8/Abstracts/v8i1r7.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Overfull graph open-in-new

Overfull graph