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## Graceful Graphs

The Graceful Graphs contest has ended. Congratulations to our winners:

Position Winner Prize
1st Tomas Rokicki United States Any item from Bathsheba Sculpture worth up to \$500
2nd Jarek Wroblewski Poland Any item from Bathsheba Sculpture worth up to \$100

These are the contestants who submitted the best solution for each problem. (Problems for which the best solution was found during the first 48 hours of the contest are omitted.)

Problem Score Solver Date Achieved

These are the best solutions received for each problem in the contest:

Problem Score Solution
2 vertices 1 (0,1)
3 vertices 3 (0,1), (1,3), (0,3)
(0,1), (0,3), (1,3)
(0,3), (0,1), (3,1)
5 vertices 9 (0,1), (0,2), (6,9), (2,6), (1,6), (0,6), (2,9), (1,9), (0,9)
(1,2), (0,2), (6,9), (2,6), (1,6), (0,6), (2,9), (1,9), (0,9)
(0,1), (7,9), (1,4), (0,4), (4,9), (1,7), (0,7), (1,9), (0,9)
(0,1), (0,2), (0,6), (0,9), (1,6), (1,9), (2,6), (2,9), (6,9)
(0,1), (7,9), (4,7), (0,4), (4,9), (1,7), (0,7), (1,9), (0,9)
(0,1), (0,4), (0,7), (0,9), (1,4), (1,7), (1,9), (4,9), (7,9)
(0,3), (0,7), (0,8), (0,9), (3,7), (3,8), (3,9), (7,8), (7,9)
7 vertices 17 (0,1), (0,2), (0,8), (0,12), (0,15), (0,17), (1,8), (1,12), (1,15), (1,17), (2,8), (2,12), (2,15), (8,12), (8,17), (12,15), (12,17)
(0,1), (13,15), (8,11), (13,17), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (0,2), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (15,17), (8,11), (13,17), (8,13), (11,17), (1,8), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (1,3), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (0,2), (14,17), (2,6), (1,6), (0,6), (10,17), (2,10), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (0,2), (0,3), (13,17), (8,13), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (12,14), (14,17), (8,12), (12,17), (8,14), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (0,2), (0,3), (0,8), (0,13), (0,17), (1,8), (1,13), (1,17), (2,8), (2,13), (2,17), (3,8), (3,13), (3,17), (8,17), (13,17)
(0,1), (11,13), (8,11), (11,15), (8,13), (11,17), (1,8), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(2,3), (1,3), (0,3), (13,17), (8,13), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(1,2), (0,2), (14,17), (10,14), (1,6), (0,6), (10,17), (6,14), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (0,2), (0,6), (0,10), (0,14), (0,17), (1,6), (1,10), (1,14), (1,17), (2,6), (2,10), (2,14), (2,17), (6,17), (10,17), (14,17)
(0,1), (15,17), (8,11), (11,15), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (8,11), (11,15), (8,13), (11,17), (1,8), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(1,2), (1,3), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (0,8), (0,11), (0,13), (0,15), (0,17), (1,8), (1,11), (1,13), (1,15), (1,17), (8,11), (8,13), (8,17), (11,13), (11,15), (11,17)
(0,1), (15,17), (1,4), (0,4), (12,17), (4,10), (10,17), (4,12), (1,10), (0,10), (1,12), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (1,4), (0,4), (10,15), (4,10), (10,17), (4,12), (1,10), (0,10), (1,12), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(0,1), (0,2), (14,17), (10,14), (1,6), (0,6), (10,17), (2,10), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (10,12), (1,4), (0,4), (10,15), (4,10), (10,17), (4,12), (1,10), (0,10), (4,15), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(0,4), (0,15), (0,1), (0,10), (0,12), (0,17), (4,15), (4,1), (4,10), (4,12), (4,17), (15,1), (15,10), (15,17), (1,10), (1,17), (10,17)
(0,1), (0,17), (0,14), (0,12), (0,8), (0,2), (1,17), (1,14), (1,12), (1,8), (17,14), (17,12), (17,8), (17,2), (14,8), (12,8), (12,2)
(1,2), (0,2), (14,17), (2,6), (1,6), (0,6), (10,17), (2,10), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(1,2), (0,2), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (13,15), (8,11), (11,15), (8,13), (11,17), (1,8), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (12,15), (8,12), (12,17), (2,8), (8,15), (0,8), (8,17), (2,12), (1,12), (0,12), (2,15), (1,15), (2,17), (1,17), (0,17)
(0,1), (0,2), (12,15), (8,12), (12,17), (2,8), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (2,15), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (12,15), (0,4), (12,17), (4,10), (10,17), (4,12), (1,10), (0,10), (4,15), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(0,1), (13,15), (8,11), (11,15), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(2,3), (0,2), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (0,2), (14,17), (8,12), (12,17), (8,14), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (10,12), (1,4), (0,4), (10,15), (4,10), (10,17), (4,12), (1,10), (0,10), (1,12), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(0,1), (0,2), (14,17), (10,14), (1,6), (0,6), (10,17), (6,14), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,4), (0,9), (0,14), (0,15), (0,16), (0,17), (4,9), (4,14), (4,15), (4,16), (4,17), (9,15), (9,16), (9,17), (14,15), (14,16), (14,17)
(0,3), (0,7), (0,11), (0,15), (0,16), (0,17), (3,7), (3,11), (3,15), (3,16), (3,17), (7,16), (7,17), (11,16), (11,17), (15,16), (15,17)
(0,1), (15,17), (1,4), (0,4), (10,15), (4,10), (10,17), (4,12), (1,10), (0,10), (4,15), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(2,3), (1,3), (0,3), (13,17), (3,8), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
(0,1), (0,2), (14,17), (8,12), (12,17), (2,8), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (12,14), (14,17), (8,12), (12,17), (8,14), (1,8), (0,8), (8,17), (2,12), (1,12), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (15,17), (1,4), (0,4), (12,17), (4,10), (10,17), (4,12), (1,10), (0,10), (4,15), (0,12), (4,17), (1,15), (0,15), (1,17), (0,17)
(1,2), (15,17), (12,15), (8,12), (12,17), (2,8), (8,15), (0,8), (8,17), (2,12), (1,12), (0,12), (2,15), (1,15), (2,17), (1,17), (0,17)
(0,1), (12,14), (14,17), (8,12), (12,17), (2,8), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (0,2), (12,15), (8,12), (12,17), (2,8), (8,15), (0,8), (8,17), (2,12), (1,12), (0,12), (2,15), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (12,15), (8,12), (12,17), (2,8), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (2,15), (1,15), (0,15), (1,17), (0,17)
(0,1), (11,13), (8,11), (13,17), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(1,2), (0,2), (14,17), (6,10), (1,6), (0,6), (10,17), (2,10), (1,10), (0,10), (6,17), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (11,13), (8,11), (11,15), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (15,17), (8,11), (13,17), (8,13), (11,17), (8,15), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(1,2), (12,14), (14,17), (8,12), (12,17), (2,8), (1,8), (0,8), (8,17), (2,12), (1,12), (0,12), (1,14), (0,14), (2,17), (1,17), (0,17)
(1,2), (12,14), (14,17), (8,12), (12,17), (8,14), (1,8), (0,8), (8,17), (2,12), (1,12), (2,14), (1,14), (0,14), (2,17), (1,17), (0,17)
(0,1), (11,13), (8,11), (13,17), (8,13), (11,17), (1,8), (0,8), (8,17), (1,11), (0,11), (1,13), (0,13), (1,15), (0,15), (1,17), (0,17)
(0,1), (1,3), (0,3), (13,17), (8,13), (2,8), (1,8), (0,8), (8,17), (3,13), (2,13), (1,13), (0,13), (3,17), (2,17), (1,17), (0,17)
11 vertices 43 (0,1), (40,42), (37,40), (1,5), (0,5), (37,43), (9,16), (1,9), (0,9), (30,40), (5,16), (30,42), (30,43), (9,23), (1,16), (0,16), (23,40), (5,23), (23,42), (23,43), (9,30), (1,23), (0,23), (16,40), (5,30), (16,42), (16,43), (9,37), (1,30), (0,30), (9,40), (5,37), (9,42), (9,43), (5,40), (1,37), (0,37), (5,43), (1,40), (0,40), (1,42), (0,42), (0,43)
(0,1), (1,3), (0,3), (34,38), (1,6), (0,6), (6,13), (34,42), (34,43), (3,13), (27,38), (1,13), (0,13), (6,20), (27,42), (27,43), (3,20), (20,38), (1,20), (0,20), (6,27), (20,42), (20,43), (3,27), (13,38), (1,27), (0,27), (6,34), (13,42), (13,43), (3,34), (6,38), (1,34), (0,34), (3,38), (6,42), (1,38), (0,38), (3,42), (3,43), (1,42), (0,42), (0,43)
(0,1), (0,3), (0,6), (0,13), (0,20), (0,27), (0,34), (0,38), (0,42), (0,43), (1,3), (1,6), (1,13), (1,20), (1,27), (1,34), (1,38), (1,42), (3,13), (3,20), (3,27), (3,34), (3,38), (3,42), (3,43), (6,13), (6,20), (6,27), (6,34), (6,38), (6,42), (13,38), (13,42), (13,43), (20,38), (20,42), (20,43), (27,38), (27,42), (27,43), (34,38), (34,42), (34,43)
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(6,401), (5,401), (4,401), (3,401), (2,401), (1,401), (0,401), (212,614), (185,588), (158,562), (104,509), (76,482), (48,455), (20,428), (212,621), (212,622), (212,623), (212,624), (212,625), (212,626), (212,627), (185,601), (158,575), (131,549), (90,509), (62,482), (34,455), (6,428), (5,428), (4,428), (3,428), (2,428), (1,428), (0,428), (185,614), (158,588), (131,562), (104,536), (76,509), (48,482), (20,455), (185,621), (185,622), (185,623), (185,624), (185,625), (185,626), (185,627), (158,601), (131,575), (104,549), (90,536), (62,509), (34,482), (6,455), (5,455), (4,455), (3,455), (2,455), (1,455), (0,455), (158,614), (131,588), (104,562), (90,549), (76,536), (48,509), (20,482), (158,621), (158,622), (158,623), (158,624), (158,625), (158,626), (158,627), (131,601), (104,575), (90,562), (76,549), (62,536), (34,509), (6,482), (5,482), (4,482), (3,482), (2,482), (1,482), (0,482), (131,614), (104,588), (90,575), (76,562), (62,549), (48,536), (20,509), (131,621), (131,622), (131,623), (131,624), (131,625), (131,626), (131,627), (104,601), (90,588), (76,575), (62,562), (48,549), (34,536), (6,509), (5,509), (4,509), (3,509), (2,509), (1,509), (0,509), (104,614), (90,601), (76,588), (62,575), (48,562), (34,549), (20,536), (104,621), (104,622), (104,623), (104,624), (104,625), (104,626), (104,627), (90,614), (76,601), (62,588), (48,575), (34,562), (20,549), (6,536), (90,621), (4,536), (90,623), (90,624), (90,625), (90,626), (90,627), (76,614), (62,601), (48,588), (34,575), (20,562), (6,549), (5,549), (4,549), (76,622), (2,549), (1,549), (0,549), (76,626), (76,627), (62,614), (48,601), (34,588), (20,575), (6,562), (5,562), (4,562), (62,621), (62,622), (62,623), (0,562), (62,625), (62,626), (62,627), (48,614), (34,601), (20,588), (6,575), (5,575), (4,575), (3,575), (48,621), (1,575), (0,575), (48,624), (48,625), (48,626), (48,627), (34,614), (20,601), (6,588), (5,588), (4,588), (3,588), (2,588), (1,588), (34,622), (34,623), (34,624), (34,625), (34,626), (34,627), (20,614), (6,601), (5,601), (4,601), (3,601), (2,601), (1,601), (0,601), (20,622), (20,623), (20,624), (20,625), (20,626), (20,627), (6,614), (5,614), (4,614), (3,614), (2,614), (1,614), (0,614), (6,621), (5,621), (4,621), (6,624), (5,624), (2,622), (3,624), (1,623), (3,626), (3,627), (0,625), (0,626), (0,627)
(2,3), (0,2), (3,6), (1,5), (0,5), (0,6), (6,13), (5,13), (4,13), (3,13), (2,13), (1,13), (26,39), (607,621), (607,622), (607,623), (607,624), (607,625), (607,626), (6,26), (5,26), (4,26), (3,26), (2,26), (1,26), (52,78), (253,280), (593,621), (593,622), (593,623), (593,624), (593,625), (593,626), (5,39), (4,39), (3,39), (2,39), (1,39), (13,52), (78,118), (496,537), (565,607), (579,622), (579,623), (579,624), (6,52), (579,626), (4,52), (3,52), (2,52), (1,52), (13,65), (65,118), (199,253), (496,551), (565,621), (565,622), (565,623), (565,624), (565,625), (4,65), (3,65), (2,65), (1,65), (0,65), (52,118), (78,145), (469,537), (496,565), (551,621), (551,622), (6,78), (5,78), (551,625), (551,626), (2,78), (1,78), (13,91), (39,118), (65,145), (361,442), (469,551), (496,579), (537,621), (537,622), (5,91), (537,624), (537,625), (537,626), (1,91), (0,91), (26,118), (52,145), (78,172), (442,537), (469,565), (496,593), (523,621), (523,622), (523,623), (523,624), (523,625), (523,626), (523,627), (13,118), (39,145), (65,172), (280,388), (442,551), (469,579), (496,607), (6,118), (5,118), (4,118), (3,118), (2,118), (1,118), (0,118), (26,145), (52,172), (78,199), (415,537), (442,565), (469,593), (496,621), (496,622), (496,623), (496,624), (496,625), (496,626), (496,627), (13,145), (39,172), (65,199), (388,523), (415,551), (442,579), (469,607), (6,145), (5,145), (4,145), (3,145), (2,145), (1,145), (0,145), (26,172), (52,199), (78,226), (388,537), (415,565), (442,593), (469,621), (469,622), (469,623), (469,624), (469,625), (469,626), (469,627), (13,172), (39,199), (65,226), (172,334), (388,551), (415,579), (442,607), (6,172), (5,172), (4,172), (3,172), (2,172), (1,172), (0,172), (26,199), (52,226), (78,253), (361,537), (388,565), (415,593), (442,621), (442,622), (442,623), (442,624), (442,625), (442,626), (442,627), (13,199), (39,226), (65,253), (280,469), (361,551), (388,579), (415,607), (6,199), (5,199), (4,199), (3,199), (2,199), (1,199), (0,199), (26,226), (52,253), (78,280), (334,537), (361,565), (388,593), (415,621), (415,622), (415,623), (415,624), (415,625), (415,626), (415,627), (13,226), (39,253), (65,280), (91,307), (334,551), (361,579), (388,607), (6,226), (5,226), (4,226), (3,226), (2,226), (1,226), (0,226), (26,253), (52,280), (78,307), (307,537), (334,565), (361,593), (388,621), (388,622), (388,623), (388,624), (388,625), (388,626), (388,627), (13,253), (39,280), (65,307), (199,442), (307,551), (334,579), (361,607), (6,253), (5,253), (4,253), (3,253), (2,253), (1,253), (0,253), (26,280), (52,307), (78,334), (280,537), (307,565), (334,593), (361,621), (361,622), (361,623), (361,624), (361,625), (361,626), (361,627), (13,280), (39,307), (65,334), (226,496), (280,551), (307,579), (334,607), (6,280), (5,280), (4,280), (3,280), (2,280), (1,280), (0,280), (26,307), (52,334), (78,361), (253,537), (280,565), (307,593), (334,621), (334,622), (334,623), (334,624), (334,625), (334,626), (334,627), (13,307), (39,334), (65,361), (226,523), (253,551), (280,579), (307,607), (6,307), (5,307), (4,307), (3,307), (2,307), (1,307), (0,307), (26,334), (52,361), (78,388), (226,537), (253,565), (280,593), (307,621), (307,622), (307,623), (307,624), (307,625), (307,626), (307,627), (13,334), (39,361), (65,388), (172,496), (226,551), (253,579), (280,607), (6,334), (5,334), (4,334), (3,334), (2,334), (1,334), (0,334), (26,361), (52,388), (78,415), (199,537), (226,565), (253,593), (280,621), (280,622), (280,623), (280,624), (280,625), (280,626), (280,627), (13,361), (39,388), (65,415), (118,469), (199,551), (226,579), (253,607), (6,361), (5,361), (4,361), (3,361), (2,361), (1,361), (0,361), (26,388), (52,415), (78,442), (172,537), (199,565), (226,593), (253,621), (253,622), (253,623), (253,624), (253,625), (253,626), (253,627), (13,388), (39,415), (65,442), (91,469), (172,551), (199,579), (226,607), (6,388), (5,388), (4,388), (3,388), (2,388), (1,388), (0,388), (26,415), (52,442), (78,469), (145,537), (172,565), (199,593), (226,621), (226,622), (226,623), (226,624), (226,625), (226,626), (226,627), (13,415), (39,442), (65,469), (118,523), (145,551), (172,579), (199,607), (6,415), (5,415), (4,415), (3,415), (2,415), (1,415), (0,415), (26,442), (52,469), (78,496), (118,537), (145,565), (172,593), (199,621), (199,622), (199,623), (199,624), (199,625), (199,626), (199,627), (13,442), (39,469), (65,496), (91,523), (118,551), (145,579), (172,607), (6,442), (5,442), (4,442), (3,442), (2,442), (1,442), (0,442), (26,469), (52,496), (78,523), (91,537), (118,565), (145,593), (172,621), (172,622), (172,623), (172,624), (172,625), (172,626), (172,627), (13,469), (39,496), (65,523), (78,537), (91,551), (118,579), (145,607), (6,469), (5,469), (4,469), (3,469), (2,469), (1,469), (0,469), (26,496), (52,523), (65,537), (78,551), (91,565), (118,593), (145,621), (145,622), (145,623), (145,624), (145,625), (145,626), (145,627), (13,496), (39,523), (52,537), (65,551), (78,565), (91,579), (118,607), (6,496), (5,496), (4,496), (3,496), (2,496), 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