28:
439:
412:
20:
450:
Analogously, the degeneracy of every triangle-free penny graph is at most two. Every such graph contains a vertex with at most two neighbors, even though it is not always possible to find this vertex on the convex hull. Based on this, one can prove that they require at most three colors, more easily
685:
It is possible to perform some computational tasks on directed penny graphs, such as testing whether one vertex can reach another, in polynomial time and substantially less than linear space, given an input representing its circles in a form allowing basic computational tasks such as testing
105:
formed from unit circles. If each vertex is represented by a point at the center of its circle, then two vertices will be adjacent if and only if their distance is the minimum distance among all pairs of vertices. Therefore, penny graphs have also been called
1498:
Bowen, Clinton; Durocher, Stephane; Löffler, Maarten; Rounds, Anika; Schulz, André; Tóth, Csaba D. (2015), "Realization of simply connected polygonal linkages and recognition of unit disk contact trees", in
Giacomo, Emilio Di;
232:
316:
401:
178:
have fewer neighbors. Counting more precisely this reduction in neighbors for boundary pennies leads to a precise bound on the number of edges in any penny graph: a penny graph with
355:
64:
1593:, Leibniz International Proceedings in Informatics (LIPIcs), vol. 212, Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 63:1–63:17,
157:; however, both upper and lower bounds are known for the size of the maximum independent set, higher than the bounds that are possible for arbitrary planar graphs.
561:
536:
515:
1506:
Graph
Drawing and Network Visualization: 23rd International Symposium, GD 2015, Los Angeles, CA, USA, September 24–26, 2015, Revised Selected Papers
1146:
185:
86:. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for every pair of
272:
1100:
674:
However, if a graph is given without geometric positions for its vertices, then testing whether it can be represented as a penny graph is
419:
Every penny graph contains a vertex with at most three neighbors. For instance, such a vertex can be found at one of the corners of the
698:(graphs that can be represented by tangencies of non-crossing circles of arbitrary radii). Because the coin graphs are the same as the
671:
of the circle centers (both of which contain the penny graph as a subgraph) and then test which edges correspond to circle tangencies.
479:
problem, in which one must find large subsets of non-overlapping regions of the plane. However, as with planar graphs more generally,
783:"A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation"
1504:
566:
251:
1553:
Graph
Drawing, 12th International Symposium, GD 2004, New York, NY, USA, September 29 - October 2, 2004, Revised Selected Papers
360:
949:
682:. Similarly, testing whether a graph can be represented as a three-dimensional mutual nearest neighbor graph is also NP-hard.
1608:
1570:
1528:
94:, arranged without overlapping on a flat surface, with a vertex for each penny and an edge for each two pennies that touch.
1351:
875:
821:
484:
321:
127:
1510:
1214:
1042:
922:
467:
in a penny graph is a subset of the pennies, no two of which touch each other. Finding maximum independent sets is
149:, but this theorem is easier to prove for penny graphs. Testing whether a graph is a penny graph, or finding its
1649:
630:
1233:(March 1975), "From rubber ropes to rolling cubes, a miscellany of refreshing problems", Mathematical Games,
1029:, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons, Inc.,
435:. However, despite their restricted structure, there exist penny graphs that do still require four colors.
1462:
Khuller, Samir; Matias, Yossi (1995), "A simple randomized sieve algorithm for the closest-pair problem",
446:
penny graph with the property that all the pennies on the convex hull touch at least three other pennies
1709:
424:
452:
1719:
990:
464:
150:
718:(graphs that can be drawn in the plane with equal-length straight edges and no edge crossings).
1714:
1204:
695:
668:
664:
171:
170:
Every vertex in a penny graph has at most six neighboring vertices; here the number six is the
123:
102:
71:
37:
1318:
1200:
865:
764:
1161:
1672:
1485:
1384:
1336:
1182:
1133:
1079:
1052:
972:
885:
844:
806:
759:
480:
476:
1624:
1066:
Kupitz, Y. S. (1994), "On the maximal number of appearances of the minimal distance among
8:
1235:
907:, Lecture Notes in Computer Science, vol. 885, Berlin: Springer-Verlag, p. 12,
711:
679:
443:
266:
1686:
714:(graphs that can be drawn with all edges having equal lengths, allowing crossings), and
543:
1614:
1445:
1427:
1360:
1294:
1275:
1252:
1248:
1109:
707:
521:
500:
432:
146:
1586:
963:
1618:
1604:
1566:
1524:
1210:
1140:
1098:(2018), "Edge bounds and degeneracy of triangle-free penny graphs and squaregraphs",
1038:
1022:
918:
871:
235:
130:
is a penny graph, although edges in different components may have different lengths.
1644:
1449:
947:(1996), "The logic engine and the realization problem for nearest neighbor graphs",
740:
1658:
1594:
1556:
1514:
1471:
1437:
1370:
1298:
1284:
1244:
1119:
1030:
958:
908:
830:
794:
782:
715:
138:
1074:, Colloq. Math. Soc. János Bolyai, vol. 63, North-Holland, pp. 217–244,
1668:
1599:
1561:
1555:, Lecture Notes in Computer Science, vol. 3383, Springer, pp. 329–339,
1519:
1481:
1380:
1332:
1178:
1129:
1075:
1048:
968:
881:
840:
802:
755:
703:
431:
require at most four colors, much more easily than the proof of the more general
415:
An optimal coloring of the 11-vertex penny graph shown above requires four colors
134:
87:
1410:
1663:
1544:
1323:
1230:
1196:
1169:
1095:
985:
944:
819:
Csizmadia, G. (1998), "On the independence number of minimum distance graphs",
781:
Cerioli, Marcia R.; Faria, Luerbio; Ferreira, Talita O.; Protti, Fábio (2011),
649:
428:
1375:
1349:
Swanepoel, Konrad J. (2002), "Independence numbers of planar contact graphs",
1273:(1994), "Approximation algorithms for NP-complete problems on planar graphs",
1703:
686:
adjacency and finding intersections of the circles with axis-parallel lines.
79:
27:
1548:
1441:
1314:
1018:
861:
573:
438:
1476:
1270:
699:
610:
was proven by
Swanepoel. In the other direction, Pach and Tóth proved that
142:
1587:"Space-efficient algorithms for reachability in directed geometric graphs"
1289:
1034:
913:
798:
663:. An alternative method with the same worst-case time is to construct the
1418:
940:
754:, MAA Notes, vol. 53, Cambridge University Press, pp. 174–194,
744:
420:
262:
258:
175:
83:
32:
1256:
596:
of the pennies that do not touch each other. By the four-color theorem,
1591:
32nd
International Symposium on Algorithms and Computation (ISAAC 2021)
1500:
1124:
835:
489:
411:
403:
bound is tight. Proving this, or finding a better bound, remains open.
240:
174:
for circles in the plane. However, the pennies on the boundary of the
1365:
1209:, Dover Books on Mathematics, Courier Corporation, pp. 177–178,
617:. As of 2013, these remained the best bounds known for this problem.
318:
and in any triangle-free penny graph the number of edges is at most
1432:
1114:
248:
What is the maximum number of edges in a triangle-free penny graph?
675:
472:
468:
154:
234:
edges. Some penny graphs, formed by arranging the pennies in a
23:
11 pennies, forming a penny graph with 11 vertices and 19 edges
122:
that links pairs of points in the plane that are each other's
19:
227:{\displaystyle \left\lfloor 3n-{\sqrt {12n-3}}\right\rfloor }
91:
311:{\displaystyle \left\lfloor 2n-2{\sqrt {n}}\right\rfloor ,}
1497:
780:
396:{\displaystyle \left\lfloor 2n-2{\sqrt {n}}\right\rfloor }
702:, all penny graphs are planar. The penny graphs are also
592:
pennies on a flat surface, there should be a subset of
584:-vertex penny graph has an independent set of at least
427:
at most three. Based on this, one can prove that their
66:
as a penny graph (the contact graph of the black disks)
625:
Constructing a penny graph from the locations of its
546:
524:
503:
363:
324:
275:
188:
40:
1206:
1162:"Triangle-free minimum distance graphs in the plane"
678:. It remains NP-hard even when the given graph is a
423:
of the circle centers. Therefore, penny graphs have
1687:"Graphs defined by matchsticks, pennies and hinges"
1260:; see problem 7, "the colored poker chips", p. 114.
1542:
903:Veltkamp, Remco C. (1994), "2.2.1 Closest pairs",
555:
530:
509:
455:that triangle-free planar graphs are 3-colorable.
395:
349:
310:
226:
58:
1547:(2004), "The Three Dimensional Logic Engine", in
648:or (with randomized time and with the use of the
1701:
1630:
1409:Dumitrescu, Adrian; Jiang, Minghui (June 2013),
1194:
1408:
939:
629:circles can be performed as an instance of the
491:
269:penny graphs whose number of edges is at least
242:
90:. The circles can be represented physically by
1642:
1589:, in Ahn, Hee-Kap; Sadakane, Kunihiko (eds.),
1513:, vol. 9411, Springer, pp. 447–459,
905:Closed Object Boundaries from Scattered Points
859:
787:RAIRO Theoretical Informatics and Applications
739:
1461:
1647:(2010), "Products of unit distance graphs",
1388:; Swanepoel's result strengthens an earlier
1145:: CS1 maint: DOI inactive as of July 2024 (
1101:Journal of Graph Algorithms and Applications
1319:"On the independence number of coin graphs"
1090:
1088:
1017:
1005:
745:"Bridges between geometry and graph theory"
620:
540:penny graph has an independent set of size
1684:
475:on penny graphs. It is an instance of the
1662:
1598:
1584:
1560:
1518:
1475:
1431:
1396:
1374:
1364:
1348:
1288:
1159:
1153:
1123:
1113:
1001:
962:
912:
834:
818:
689:
1313:
1094:
1085:
984:
902:
437:
410:
26:
18:
1309:
1307:
1229:
1223:
978:
735:
733:
731:
694:Penny graphs are a special case of the
567:(more unsolved problems in mathematics)
252:(more unsolved problems in mathematics)
1702:
1491:
1188:
1065:
935:
933:
867:Research Problems in Discrete Geometry
776:
774:
772:
1352:Discrete & Computational Geometry
1342:
1269:
1263:
1059:
898:
896:
894:
855:
853:
822:Discrete & Computational Geometry
238:, have exactly this number of edges.
16:Graph formed by touching unit circles
1685:Feuilloley, Laurent (May 29, 2019),
1636:
1578:
1536:
1455:
1402:
1304:
728:
485:polynomial-time approximation scheme
1426:(2), New York, NY, US: ACM: 80–87,
1011:
930:
870:, New York: Springer, p. 228,
812:
769:
458:
451:than the proof of the more general
350:{\displaystyle 2n-1.65{\sqrt {n}}.}
97:Penny graphs have also been called
13:
1678:
1585:Bhore, Sujoy; Jain, Rahul (2021),
1411:"Computational Geometry Column 56"
1249:10.1038/scientificamerican0375-112
988:(1974), "Lösung zu Problem 664A",
891:
850:
471:for arbitrary graphs, and remains
165:
14:
1731:
1511:Lecture Notes in Computer Science
1072:Intuitive Geometry (Szeged, 1991)
1056:; see Theorem 13.12, p. 211.
750:, in Gorini, Catherine A. (ed.),
588:vertices. That is, if we place
492:Unsolved problem in mathematics
357:Swanepoel conjectured that the
243:Unsolved problem in mathematics
1631:Hartsfield & Ringel (2013)
860:Brass, Peter; Moser, William;
631:closest pair of points problem
257:By arranging the pennies in a
145:more generally, they obey the
1:
1633:, Theorem 8.4.2, p. 173.
1160:Swanepoel, Konrad J. (2009),
964:10.1016/S0304-3975(97)84223-5
721:
576:asked for the largest number
160:
120:mutual nearest neighbor graph
1600:10.4230/LIPIcs.ISAAC.2021.63
1562:10.1007/978-3-540-31843-9_33
1520:10.1007/978-3-319-27261-0_37
950:Theoretical Computer Science
261:, or in the form of certain
7:
1464:Information and Computation
406:
10:
1736:
1664:10.1016/j.disc.2009.11.035
1376:10.1007/s00454-002-2897-y
1006:Pach & Agarwal (1995)
633:, taking worst-case time
603:, and the improved bound
59:{\displaystyle H_{3}^{5}}
743:; Randić, Milan (2000),
621:Computational complexity
112:smallest-distance graphs
1442:10.1145/2491533.2491550
1128:(inactive 2024-07-29),
991:Elemente der Mathematik
465:maximum independent set
151:maximum independent set
133:Every penny graph is a
108:minimum-distance graphs
101:, because they are the
1477:10.1006/inco.1995.1049
1070:points in the plane",
1027:Combinatorial Geometry
690:Related graph families
669:nearest neighbor graph
665:Delaunay triangulation
557:
532:
511:
447:
416:
397:
351:
312:
228:
72:geometric graph theory
67:
60:
24:
1317:; Tóth, Géza (1996),
1290:10.1145/174644.174650
1035:10.1002/9781118033203
914:10.1007/3-540-58808-6
558:
533:
512:
441:
414:
398:
352:
313:
229:
182:vertices has at most
61:
30:
22:
1650:Discrete Mathematics
712:unit distance graphs
544:
522:
501:
497:What is the largest
477:maximum disjoint set
361:
322:
273:
186:
116:closest-pairs graphs
38:
1543:Kitching, Matthew;
1236:Scientific American
799:10.1051/ita/2011106
708:intersection graphs
128:connected component
55:
1276:Journal of the ACM
1195:Hartsfield, Nora;
1125:10.7155/jgaa.00463
1023:Agarwal, Pankaj K.
836:10.1007/PL00009381
710:of unit circles),
556:{\displaystyle cn}
553:
528:
507:
487:for this problem.
453:Grötzsch's theorem
448:
433:four-color theorem
417:
393:
347:
308:
224:
147:four color theorem
118:. Similarly, in a
68:
56:
41:
25:
1657:(12): 1783–1792,
1610:978-3-95977-214-3
1572:978-3-540-24528-5
1530:978-3-319-27260-3
763:; see especially
716:matchstick graphs
531:{\displaystyle n}
510:{\displaystyle c}
481:Baker's technique
386:
342:
298:
217:
124:nearest neighbors
1727:
1710:Geometric graphs
1694:
1693:
1682:
1676:
1675:
1666:
1640:
1634:
1628:
1622:
1621:
1602:
1582:
1576:
1575:
1564:
1540:
1534:
1533:
1522:
1495:
1489:
1488:
1479:
1459:
1453:
1452:
1435:
1415:
1406:
1400:
1397:Csizmadia (1998)
1394:
1387:
1378:
1368:
1346:
1340:
1339:
1311:
1302:
1301:
1292:
1267:
1261:
1259:
1227:
1221:
1219:
1192:
1186:
1185:
1166:
1157:
1151:
1150:
1144:
1136:
1127:
1117:
1092:
1083:
1082:
1069:
1063:
1057:
1055:
1015:
1009:
1002:Swanepoel (2009)
999:
982:
976:
975:
966:
937:
928:
927:
916:
900:
889:
888:
877:978-0387-23815-9
857:
848:
847:
838:
816:
810:
809:
778:
767:
762:
752:Geometry at Work
749:
737:
704:unit disk graphs
662:
652:) expected time
647:
628:
616:
609:
602:
595:
591:
587:
583:
580:such that every
579:
562:
560:
559:
554:
539:
537:
535:
534:
529:
517:such that every
516:
514:
513:
508:
493:
459:Independent sets
402:
400:
399:
394:
392:
388:
387:
382:
356:
354:
353:
348:
343:
338:
317:
315:
314:
309:
304:
300:
299:
294:
244:
233:
231:
230:
225:
223:
219:
218:
204:
181:
139:matchstick graph
99:unit coin graphs
65:
63:
62:
57:
54:
49:
1735:
1734:
1730:
1729:
1728:
1726:
1725:
1724:
1700:
1699:
1698:
1697:
1683:
1679:
1645:Pisanski, Tomaž
1643:Horvat, Boris;
1641:
1637:
1629:
1625:
1611:
1583:
1579:
1573:
1545:Whitesides, Sue
1541:
1537:
1531:
1496:
1492:
1460:
1456:
1413:
1407:
1403:
1389:
1347:
1343:
1312:
1305:
1268:
1264:
1231:Gardner, Martin
1228:
1224:
1217:
1201:"Problem 8.4.8"
1197:Ringel, Gerhard
1193:
1189:
1164:
1158:
1154:
1138:
1137:
1096:Eppstein, David
1093:
1086:
1067:
1064:
1060:
1045:
1016:
1012:
983:
979:
945:Whitesides, Sue
938:
931:
925:
901:
892:
878:
858:
851:
817:
813:
779:
770:
747:
741:Pisanski, Tomaž
738:
729:
724:
692:
653:
634:
626:
623:
615:≤ 5/16 = 0.3125
611:
604:
597:
593:
589:
585:
581:
577:
570:
569:
564:
545:
542:
541:
523:
520:
519:
518:
502:
499:
498:
495:
461:
429:graph colorings
409:
381:
368:
364:
362:
359:
358:
337:
323:
320:
319:
293:
280:
276:
274:
271:
270:
265:, one can form
255:
254:
249:
246:
236:triangular grid
203:
193:
189:
187:
184:
183:
179:
168:
166:Number of edges
163:
135:unit disk graph
88:tangent circles
50:
45:
39:
36:
35:
17:
12:
11:
5:
1733:
1723:
1722:
1720:Circle packing
1717:
1712:
1696:
1695:
1691:Discrete notes
1677:
1635:
1623:
1609:
1577:
1571:
1535:
1529:
1490:
1454:
1401:
1393:≥ 9/35 ≈ 0.257
1359:(4): 649–670,
1341:
1324:Geombinatorics
1303:
1283:(1): 153–180,
1262:
1243:(3): 112–117,
1222:
1215:
1187:
1170:Geombinatorics
1152:
1108:(3): 483–499,
1084:
1058:
1043:
1010:
1000:. As cited by
977:
929:
923:
890:
876:
849:
829:(2): 179–187,
811:
793:(3): 331–346,
768:
726:
725:
723:
720:
691:
688:
650:floor function
622:
619:
608:≥ 8/31 ≈ 0.258
565:
552:
549:
527:
506:
496:
490:
460:
457:
408:
405:
391:
385:
380:
377:
374:
371:
367:
346:
341:
336:
333:
330:
327:
307:
303:
297:
292:
289:
286:
283:
279:
250:
247:
241:
222:
216:
213:
210:
207:
202:
199:
196:
192:
172:kissing number
167:
164:
162:
159:
53:
48:
44:
15:
9:
6:
4:
3:
2:
1732:
1721:
1718:
1716:
1715:Planar graphs
1713:
1711:
1708:
1707:
1705:
1692:
1688:
1681:
1674:
1670:
1665:
1660:
1656:
1652:
1651:
1646:
1639:
1632:
1627:
1620:
1616:
1612:
1606:
1601:
1596:
1592:
1588:
1581:
1574:
1568:
1563:
1558:
1554:
1550:
1546:
1539:
1532:
1526:
1521:
1516:
1512:
1508:
1507:
1502:
1494:
1487:
1483:
1478:
1473:
1469:
1465:
1458:
1451:
1447:
1443:
1439:
1434:
1429:
1425:
1421:
1420:
1412:
1405:
1398:
1392:
1386:
1382:
1377:
1372:
1367:
1362:
1358:
1354:
1353:
1345:
1338:
1334:
1330:
1326:
1325:
1320:
1316:
1310:
1308:
1300:
1296:
1291:
1286:
1282:
1278:
1277:
1272:
1266:
1258:
1254:
1250:
1246:
1242:
1238:
1237:
1232:
1226:
1218:
1216:9780486315522
1212:
1208:
1207:
1202:
1198:
1191:
1184:
1180:
1176:
1172:
1171:
1163:
1156:
1148:
1142:
1135:
1131:
1126:
1121:
1116:
1111:
1107:
1103:
1102:
1097:
1091:
1089:
1081:
1077:
1073:
1062:
1054:
1050:
1046:
1044:0-471-58890-3
1040:
1036:
1032:
1028:
1024:
1020:
1014:
1007:
1003:
997:
993:
992:
987:
981:
974:
970:
965:
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1549:Pach, János
1501:Lubiw, Anna
1419:SIGACT News
1315:Pach, János
1019:Pach, János
862:Pach, János
765:p. 176
696:coin graphs
483:provides a
421:convex hull
259:square grid
176:convex hull
103:coin graphs
76:penny graph
33:Hanoi graph
1704:Categories
1433:cs/9908007
1115:1708.05152
722:References
574:Paul Erdős
425:degeneracy
161:Properties
1619:244731943
1395:bound of
1271:Baker, B.
572:In 1983,
376:−
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