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Line graph of a hypergraph
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<h2 id="line-graphs-of-k-uniform-hypergraphs-k-3">Line graphs of <i>k</i>-uniform hypergraphs, <i>k</i> ≥ 3</h2> <p>Beineke<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> characterized line graphs of graphs by a list of 9 <a href="/facts/Forbidden_induced_subgraph/50Kex6JU">forbidden induced subgraphs</a>. (See the article on <a href="/facts/Line_graph/RWNxSy5j">line graphs</a>.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any <i>k</i> ≥ 3, and Lovász<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> showed there is no such characterization by a finite list if <i>k</i> = 3. </p><p>Krausz<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> characterized line graphs of graphs in terms of <a href="/facts/Clique_(graph_theory)/XK9ppLz5">clique</a> covers. (See <a href="/facts/Line_graph/RWNxSy5j">Line Graphs</a>.) A global characterization of Krausz type for the line graphs of <i>k</i>-uniform hypergraphs for any <i>k</i> ≥ 3 was given by Berge<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="line-graphs-of-k-uniform-linear-hypergraphs-k-3">Line graphs of <i>k</i>-uniform linear hypergraphs, <i>k</i> ≥ 3</h2> <p>A global characterization of Krausz type for the line graphs of <i>k</i>-uniform linear hypergraphs for any <i>k</i> ≥ 3 was given by Naik, Rao, Shrikhande, and Singhi.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. Metelsky|l and Tyshkevich<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> and Jacobson, Kézdy, and Lehel<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> improved this bound to 19. At last Skums, Suzdal', and Tyshkevich<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> reduced this bound to 16. Metelsky and Tyshkevich<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> also proved that, if <i>k</i> > 3, no such finite list exists for linear <i>k</i>-uniform hypergraphs, no matter what lower bound is placed on the degree. </p><p>The difficulty in finding a characterization of linear <i>k</i>-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for <i>m</i> > 0, consider a chain of <i>m</i> <a href="/facts/Diamond_graph/8awrY0xS">diamond graphs</a> such that the consecutive diamonds share vertices of degree two. For <i>k</i> ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs of Naik, Rao, Shrikhande, and Singhi<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke's of line graphs of graphs. </p> <p>There are some interesting characterizations available for line graphs of linear <i>k</i>-uniform hypergraphs due to various authors<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least <i>k</i>3-2<i>k</i>2+1 in Naik, Rao, Shrikhande, and Singhi<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> is reduced to 2<i>k</i>2-3<i>k</i>+1 in Jacobson, Kézdy, and Lehel<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> and Zverovich<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> to characterize line graphs of <i>k</i>-uniform linear hypergraphs for any <i>k</i> ≥ 3. </p><p>The complexity of recognizing line graphs of linear <i>k</i>-uniform hypergraphs without any constraint on minimum degree (or minimum edge-degree) is not known. For <i>k</i> = 3 and minimum degree at least 19, recognition is possible in polynomial time.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> Skums, Suzdal', and Tyshkevich<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> reduced the minimum degree to 10. </p><p>There are many interesting open problems and conjectures in Naik et al., Jacoboson et al., Metelsky et al. and Zverovich. </p> <h2 id="disjointness-graph">Disjointness graph</h2> <p>The disjointness graph of a hypergraph <i>H</i>, denoted D(<i>H</i>), is the graph whose vertex set is the set of the hyperedges of <i>H</i>, with two vertices adjacent in D(<i>H</i>) when their corresponding hyperedges are <i>disjoint</i> in <i>H</i>.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> In other words, D(<i>H</i>) is the <a href="/facts/Complement_graph/IdHUPFr3">complement graph</a> of L(<i>H</i>). A <a href="/facts/Clique_(graph_theory)/XK9ppLz5">clique</a> in D(<i>H</i>) corresponds to an independent set in L(<i>H</i>), and vice versa. </p> <ul><li>Beineke, L. W. (1968), "On derived graphs and digraphs", in Sachs, H.; Voss, H.; Walther, H. (eds.), <i>Beitrage zur Graphentheorie</i>, Leipzig: Teubner, pp. 17–23.</li> <li><a href="/facts/Claude_Berge/VPsApqnu">Berge, C.</a> (1989), <i>Hypergraphs: Combinatorics of Finite Sets</i>, Amsterdam: North-Holland, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1013569">1013569</a>. Translated from the French.</li> <li>Bermond, J. C.; Heydemann, M. C.; Sotteau, D. (1977), <a href="https://hal.inria.fr/hal-02360671/file/21-BHS77-L%28H%29.pdf">"Line graphs of hypergraphs I"</a> (PDF), <i><a href="/facts/Discrete_Mathematics_(journal)/K4e6z8wQ">Discrete Mathematics</a></i>, 18 (3): 235–241, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0012-365X%2877%2990127-3">10.1016/0012-365X(77)90127-3</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463003">0463003</a>.</li></ul> <ul><li>Heydemann, M. C.; Sotteau, D. (1976), "Line graphs of hypergraphs II", <i>Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976)</i>, Colloq. Math. Soc. J. Bolyai, vol. 18, pp. 567–582, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0519291">0519291</a>.</li></ul> <ul><li>Krausz, J. (1943), "Démonstration nouvelle d'une théorème de Whitney sur les réseaux", <i>Mat. Fiz. Lapok</i>, 50: 75–85, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0018403">0018403</a>. (In Hungarian, with French abstract.)</li></ul> <ul><li><a href="/facts/L%C3%A1szl%C3%B3_Lov%C3%A1sz/X0VB38Ee">Lovász, L.</a> (1977), "Problem 9", <i>Beiträge zur Graphentheorie und deren Anwendungen</i>, Vorgetragen auf dem Internationalen Kolloquium in Oberhof (DDR), p. 313.</li></ul> <ul><li>Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", <i><a href="/facts/Graphs_and_Combinatorics/kQJQ9uUy">Graphs and Combinatorics</a></i>, 13 (4): 359–367, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF03353014">10.1007/BF03353014</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1485929">1485929</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:9173731">9173731</a>.</li></ul> <ul><li>Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of linear 3-uniform hypergraphs", <i>Journal of Graph Theory</i>, 25 (4): 243–251, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2F%28SICI%291097-0118%28199708%2925%3A4%3C243%3A%3AAID-JGT1%3E3.0.CO%3B2-K">10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1459889">1459889</a>.</li></ul> <ul><li>Naik, Ranjan N.; Rao, S. B.; <a href="/facts/S._S._Shrikhande/hnRScWY1">Shrikhande, S. S.</a>; Singhi, N. M. (1980), "Intersection graphs of <i>k</i>-uniform hypergraphs", <i>Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978)</i>, Annals of Discrete Mathematics, vol. 6, pp. 275–279, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0593539">0593539</a>.</li></ul> <ul><li>Naik, Ranjan N.; Rao, S. B.; <a href="/facts/S._S._Shrikhande/hnRScWY1">Shrikhande, S. S.</a>; Singhi, N. M. (1982), "Intersection graphs of <i>k</i>-uniform linear hypergraphs", <i>European Journal of Combinatorics</i>, 3 (2): 159–172, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fs0195-6698%2882%2980029-2">10.1016/s0195-6698(82)80029-2</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0670849">0670849</a>.</li></ul> <ul><li>Skums, P. V.; Suzdal', S. V.; Tyshkevich, R. I. (2009), "Edge intersection of linear 3-uniform hypergraphs", <i>Discrete Mathematics</i>, 309: 3500–3517, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.disc.2007.12.082">10.1016/j.disc.2007.12.082</a>.</li></ul> <ul><li>Zverovich, Igor E. (2004), "A solution to a problem of Jacobson, Kézdy and Lehel", <i><a href="/facts/Graphs_and_Combinatorics/kQJQ9uUy">Graphs and Combinatorics</a></i>, 20 (4): 571–577, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00373-004-0572-1">10.1007/s00373-004-0572-1</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2108401">2108401</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:33662052">33662052</a>.</li></ul> <ul><li>Voloshin, Vitaly I. (2009), <i>Introduction to Graph and Hypergraph Theory</i>, New York: <a href="/facts/Nova_Science_Publishers,_Inc./xwUGA1CU">Nova Science Publishers, Inc.</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2514872">2514872</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>(Berge 1989) - Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, Amsterdam: North-Holland, MR 1013569 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1013569" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1013569</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Beineke (1968) - Beineke, L. W. (1968), "On derived graphs and digraphs", in Sachs, H.; Voss, H.; Walther, H. (eds.), Beitrage zur Graphentheorie, Leipzig: Teubner, pp. 17–23 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lovász (1977) - Lovász, L. (1977), "Problem 9", Beiträge zur Graphentheorie und deren Anwendungen, Vorgetragen auf dem Internationalen Kolloquium in Oberhof (DDR), p. 313 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Krausz (1943) - Krausz, J. (1943), "Démonstration nouvelle d'une théorème de Whitney sur les réseaux", Mat. Fiz. Lapok, 50: 75–85, MR 0018403 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0018403" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0018403</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Berge (1989) - Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, Amsterdam: North-Holland, MR 1013569 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1013569" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1013569</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Naik et al. (1980) - Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), "Intersection graphs of k-uniform hypergraphs", Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics, vol. 6, pp. 275–279, MR 0593539 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0593539" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0593539</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Metelsky & Tyshkevich (1997) - Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of linear 3-uniform hypergraphs", Journal of Graph Theory, 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR 1459889 <a href="https://doi.org/10.1002%2F%28SICI%291097-0118%28199708%2925%3A4%3C243%3A%3AAID-JGT1%3E3.0.CO%3B2-K" target="_blank">https://doi.org/10.1002%2F%28SICI%291097-0118%28199708%2925%3A4%3C243%3A%3AAID-JGT1%3E3.0.CO%3B2-K</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Jacobson, Kézdy & Lehel (1997) - Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", Graphs and Combinatorics, 13 (4): 359–367, doi:10.1007/BF03353014, MR 1485929, S2CID 9173731 <a href="https://doi.org/10.1007%2FBF03353014" target="_blank">https://doi.org/10.1007%2FBF03353014</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Skums, Suzdal' & Tyshkevich (2009) - Skums, P. V.; Suzdal', S. V.; Tyshkevich, R. I. (2009), "Edge intersection of linear 3-uniform hypergraphs", Discrete Mathematics, 309: 3500–3517, doi:10.1016/j.disc.2007.12.082 <a href="https://doi.org/10.1016%2Fj.disc.2007.12.082" target="_blank">https://doi.org/10.1016%2Fj.disc.2007.12.082</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Metelsky & Tyshkevich (1997) - Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of linear 3-uniform hypergraphs", Journal of Graph Theory, 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR 1459889 <a href="https://doi.org/10.1002%2F%28SICI%291097-0118%28199708%2925%3A4%3C243%3A%3AAID-JGT1%3E3.0.CO%3B2-K" target="_blank">https://doi.org/10.1002%2F%28SICI%291097-0118%28199708%2925%3A4%3C243%3A%3AAID-JGT1%3E3.0.CO%3B2-K</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Naik et al. (1980), Naik et al. (1982) - Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), "Intersection graphs of k-uniform hypergraphs", Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics, vol. 6, pp. 275–279, MR 0593539 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0593539" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0593539</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Naik et al. (1980), Naik et al. (1982), Jacobson, Kézdy & Lehel 1997, Metelsky & Tyshkevich 1997, and Zverovich 2004 - Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), "Intersection graphs of k-uniform hypergraphs", Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics, vol. 6, pp. 275–279, MR 0593539 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0593539" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0593539</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Naik et al. (1980) - Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), "Intersection graphs of k-uniform hypergraphs", Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics, vol. 6, pp. 275–279, MR 0593539 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0593539" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0593539</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Jacobson, Kézdy & Lehel (1997) - Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", Graphs and Combinatorics, 13 (4): 359–367, doi:10.1007/BF03353014, MR 1485929, S2CID 9173731 <a href="https://doi.org/10.1007%2FBF03353014" target="_blank">https://doi.org/10.1007%2FBF03353014</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Zverovich (2004) - Zverovich, Igor E. (2004), "A solution to a problem of Jacobson, Kézdy and Lehel", Graphs and Combinatorics, 20 (4): 571–577, doi:10.1007/s00373-004-0572-1, MR 2108401, S2CID 33662052 <a href="https://doi.org/10.1007%2Fs00373-004-0572-1" target="_blank">https://doi.org/10.1007%2Fs00373-004-0572-1</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Jacobson, Kézdy & Lehel 1997 and Metelsky & Tyshkevich 1997 - Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", Graphs and Combinatorics, 13 (4): 359–367, doi:10.1007/BF03353014, MR 1485929, S2CID 9173731 <a href="https://doi.org/10.1007%2FBF03353014" target="_blank">https://doi.org/10.1007%2FBF03353014</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Skums, Suzdal' & Tyshkevich (2009) - Skums, P. V.; Suzdal', S. V.; Tyshkevich, R. I. (2009), "Edge intersection of linear 3-uniform hypergraphs", Discrete Mathematics, 309: 3500–3517, doi:10.1016/j.disc.2007.12.082 <a href="https://doi.org/10.1016%2Fj.disc.2007.12.082" target="_blank">https://doi.org/10.1016%2Fj.disc.2007.12.082</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> </ol>