Abstract
Let \(\lambda K_v\) be the complete graph on v vertices in which each pair of vertices is joined by exactly \(\lambda \) edges. An m-cycle system of \(\lambda K_v\) is a collection \({\mathscr {C}}\) of cycles of length m whose edges partition the edges of \(\lambda K_v\). An m-cycle system \({\mathscr {C}}\) of \(\lambda K_v\) is said to be resolvable if the m-cycles in \({\mathscr {C}}\) can be partitioned into parallel classes \({\mathscr {R}}= \{R_1,R_2,\ldots ,R_{\lambda (v-1)/2}\}\) and \({\mathscr {C}}\) is denoted by \((v,m,\lambda )\)-RCS, \({\mathscr {R}}\) is called a resolution. If a \((v,m,\lambda )\)-RCS has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted by \((v, m,\lambda )\)-DRCS. In this paper, applying direct constructions and recursive constructions, we show that a (v, 4, 2)-DRCS exists if and only if \( v \equiv 0\pmod 4\).
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References
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Acknowledgements
The authors are grateful to the referees for their careful reading of the original version of this paper, their detailed comments and the suggestions that much improved the quality of this paper. A portion of this research was carried out while the first author was visiting the University of Tsukuba. He wishes to express his gratitude to the Faculty of Engineering, Information and Systems for their hospitality.
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Research supported by the National Natural Science Foundation of China under Grant No. 11371207.
Appendices
Appendix 1
Here, using the method described in Construction 11, we provide starters and adders for (4, 2)-cycle frames of types \(2^u\), \(3^u\), \(4^u\) and \(8^u\).
(1) Starter blocks and adders for (4, 2)-cycle frames of type \(2^{u}\) with \(u\in \{7,9,11,15,19\}\)
u | Starter | Adder | Starter + adder | Starter | Adder | Starter + adder |
|---|---|---|---|---|---|---|
7 | (1, 2, 12, 3) | 3 | (4, 5, 1, 6) | (4, 9, 13, 10) | 13 | (3, 8, 12, 9) |
(5, 6, 8, 11) | 5 | (10, 11, 13, 2) | ||||
9 | (1, 2, 3, 13) | 4 | (5, 6, 7, 17) | (4, 6, 11, 8) | 8 | (12, 14, 1, 16) |
(5, 10, 16, 12) | 10 | (15, 2, 8, 4) | (7, 14, 17, 15) | 14 | (3, 10, 13, 11) | |
11 | (1, 2, 3, 5) | 5 | (6, 7, 8, 10) | (4, 6, 14, 19) | 17 | (21, 1, 9, 14) |
(7, 13, 10, 20) | 6 | (13, 19, 16, 4) | (8, 18, 15, 21) | 19 | (5, 15, 12, 18) | |
(9, 16, 12, 17) | 8 | (17, 2, 20, 25) | ||||
15 | (4, 5, 13, 22) | 14 | (18, 19, 27, 6) | (3, 8, 17, 23) | 21 | (24, 29, 8, 14) |
(9, 10, 14, 20) | 12 | (21, 22, 26, 2) | (7, 11, 19, 2) | 9 | (16, 20, 28, 11) | |
(1, 18, 21, 24) | 16 | (17, 4, 7, 10) | (6, 25, 27, 29) | 6 | (12, 1, 3, 5) | |
(12, 26, 16, 28) | 27 | (9, 23, 13, 25) | ||||
19 | (18, 33, 24, 36) | 12 | (30, 7, 36, 10) | (1, 2, 3, 5) | 32 | (33, 34, 35, 37) |
(4, 6, 9, 12) | 9 | (13, 15, 18, 21) | (7, 11, 16, 21) | 1 | (8, 12, 17, 22) | |
(8, 14, 23, 29) | 35 | (5, 11, 20, 26) | (10, 26, 15, 28) | 37 | (9, 25, 14, 27) | |
(13, 34, 27, 35) | 27 | (2, 23, 16, 24) | (17, 30, 20, 31) | 11 | (28, 3, 31, 4) | |
(22, 32, 25, 37) | 7 | (29, 1, 32, 6) |
(2) Starter blocks and adders for (4, 2)-cycle frames of type \(3^{u}\) with \(u\in \{9,13\}\)
u | Starter | Adder | Starter + adder | Starter | Adder | Starter + adder |
|---|---|---|---|---|---|---|
9 | (4, 5, 21, 17) | 8 | (12, 13, 2, 25) | (2, 15, 20, 12) | 26 | (1, 14, 19, 11) |
(7, 8, 11, 13) | 24 | (4, 5, 8, 10) | (3, 23, 6, 14) | 20 | (23, 16, 26, 7) | |
(1, 16, 10, 25) | 5 | (6, 21, 15, 3) | (19, 24, 22, 26) | 25 | (17, 22, 20, 24) | |
13 | (4, 14, 19, 29) | 11 | (15, 25, 30, 1) | (6, 8, 16, 36) | 2 | (8, 10, 18, 38) |
(1, 28, 5, 33) | 23 | (24, 12, 28, 17) | (7, 25, 22, 21) | 37 | (5, 23, 20, 19) | |
(12, 18, 35, 20) | 15 | (27, 33, 11, 35) | (3, 38, 17, 23) | 38 | (2, 37, 16, 22) | |
(2, 9, 31, 30) | 12 | (14, 21, 4, 3) | (10, 34, 32, 37) | 36 | (7, 31, 29, 34) | |
(11, 15, 24, 27) | 21 | (32, 36, 6, 9) |
(3) Starter blocks and adders for (4, 2)-cycle frames of type \(4^{u}\) with \(u\in \{5,6,\ldots ,10\}\)
u | Starter | Adder | Starter + adder | Starter | Adder | Starter + adder |
|---|---|---|---|---|---|---|
5 | (1, 2, 4, 17) | 2 | (3, 4, 6, 19) | (3, 6, 12, 11) | 6 | (9, 12, 18, 17) |
(7, 14, 8, 19) | 14 | (1, 8, 2, 13) | (9, 13, 16, 18) | 18 | (7, 11, 14, 16) | |
6 | (11, 15, 22, 21) | 4 | (15, 19, 2, 1) | (1, 3, 14, 9) | 8 | (9, 11, 22, 17) |
(2, 13, 20, 17) | 14 | (16, 3, 10, 7) | (4, 5, 19, 23) | 9 | (13, 14, 4, 8) | |
(7, 10, 8, 16) | 13 | (20, 23, 21, 5) | ||||
7 | (4, 6, 22, 26) | 4 | (8, 10, 26, 2) | (2, 25, 15, 18) | 16 | (18, 13, 3, 6) |
(8, 17, 20, 9) | 3 | (11, 20, 23, 12) | (1, 19, 11, 24) | 8 | (9, 27, 19, 4) | |
(3, 5, 13, 12) | 12 | (15, 17, 25, 24) | (10, 16, 27, 23) | 6 | (16, 22, 5, 1) | |
8 | (4, 10, 11, 26) | 3 | (7, 13, 14, 29) | (1, 20, 17, 3) | 27 | (28, 15, 12, 30) |
(7, 9, 13, 27) | 28 | (3, 5, 9, 23) | (2, 28, 23, 12) | 31 | (1, 27, 22, 11) | |
(5, 18, 21, 25) | 13 | (18, 31, 2, 6) | (6, 29, 22, 31) | 20 | (26, 17, 10, 19) | |
(14, 15, 30, 19) | 6 | (20, 21, 4, 25) | ||||
9 | (14, 19, 21, 33) | 5 | (19, 24, 26, 2) | (6, 29, 28, 20) | 22 | (28, 15, 14, 6) |
(7, 11, 25, 22) | 10 | (17, 21, 35, 32) | (1, 17, 23, 8) | 29 | (30, 10, 16, 1) | |
(13, 16, 3, 2) | 31 | (8, 11, 34, 33) | (4, 12, 24, 30) | 19 | (23, 31, 7, 13) | |
(5, 15, 32, 34) | 7 | (12, 22, 3, 5) | (10, 26, 31, 35) | 30 | (4, 20, 25, 29) | |
10 | (9, 12, 11, 22) | 4 | (13, 16, 15, 26) | (2, 36, 8, 6) | 33 | (35, 29, 1, 39) |
(17, 38, 32, 39) | 6 | (23, 4, 38, 5) | (16, 34, 37, 25) | 27 | (3, 21, 24, 12) | |
(1, 5, 3, 18) | 31 | (32, 36, 34, 9) | (4, 29, 15, 31) | 2 | (6, 31, 17, 33) | |
(7, 21, 13, 24) | 1 | (8, 22, 14, 25) | (14, 23, 28, 33) | 14 | (28, 37, 42, 7) | |
(19, 26, 27, 35) | 32 | (11, 18, 19, 27) |
(4) Starter blocks and adders for (4, 2)-cycle frames of type \(8^{u}\) with \(u\in \{4,5,6\}\)
u | Starter | Adder | Starter + adder | Starter | Adder | Starter + adder |
|---|---|---|---|---|---|---|
4 | (13, 14, 25, 30) | 1 | (14, 15, 26, 31) | (11, 17, 27, 29) | 26 | (5, 11, 21, 23) |
(3, 10, 7, 26) | 31 | (2, 9, 6, 25) | (2, 23, 22, 15) | 27 | (29, 18, 17, 10) | |
(6, 9, 18, 1) | 21 | (27, 30, 7, 22) | (5, 19, 21, 31) | 14 | (19, 1, 3, 13) | |
5 | (11, 17, 21, 33) | 1 | (12, 18, 22, 34) | (9, 26, 19, 36) | 37 | (6, 23, 16, 33) |
(8, 14, 38, 27) | 34 | (2, 8, 32, 21) | (18, 29, 28, 16) | 8 | (26, 37, 36, 24) | |
(1, 4, 31, 34) | 13 | (14, 17, 4, 7) | (2, 6, 32, 24) | 7 | (9, 13, 39, 31) | |
(3, 12, 13, 22) | 16 | (19, 28, 29, 38) | (7, 23, 37, 39) | 4 | (11, 27, 1, 3) | |
6 | (13, 21, 44, 47) | 2 | (15, 23, 46, 1) | (11, 28, 38, 31) | 3 | (14, 31, 41, 34) |
(3, 14, 23, 19) | 14 | (17, 28, 37, 33) | (4, 5, 45, 17) | 47 | (3, 4, 44, 16) | |
(25, 27, 46, 41) | 34 | (11, 13, 32, 27) | (8, 22, 26, 39) | 17 | (25, 39, 43, 8) | |
(1, 16, 37, 40) | 37 | (38, 5, 26, 29) | (2, 9, 7, 29) | 38 | (40, 47, 45, 19) | |
(10, 32, 43, 33) | 25 | (35, 9, 20, 10) | (15, 20, 35, 34) | 35 | (2, 7, 22, 21) |
Appendix 2
Here, using the method described in Construction 14, we provide intransitive starters (S, C, R) and adders A for (4, 2)-cycle frames of types \(4^{10}4^1\), \(8^64^1\), \(8^74^1\), \(8^78^1\), \(8^812^1\), \(8^98^1\), \(8^912^1\), \(8^{13}8^1\), \(12^78^1\), \(12^712^1\) and \(12^716^1\).
(1) (4, 2)-cycle frame of type \(4^{10}4^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(5, 11, 16, 22) | ||
R | ||
(13, 24, 15, 26) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(34, 18, 14, 38) | 7 | (1, 25, 21, 5) |
(4, 7, 9, 23) | 25 | (29, 32, 34, 8) |
(12, 24, 17, 35) | 32 | (4, 16, 9, 27) |
(15, 27, 29, 8) | 4 | (19, 31, 33, 12) |
(2, 28, 19, 37) | 9 | (11, 37, 28, 6) |
(1, 33, 32, \({\infty }_{1}\)) | 6 | (7, 39, 38, \({\infty }_{1}\)) |
(3, 6, 31, \({\infty }_{2}\)) | 11 | (14, 17, 2, \({\infty }_{2}\)) |
(13, 21, 36, \({\infty }_{3}\)) | 22 | (35, 3, 18, \({\infty }_{3}\)) |
(25, 26, 39, \({\infty }_{4}\)) | 37 | (22, 23, 36, \({\infty }_{4}\)) |
(2) (4, 2)-cycle frame of type \(8^{6}4^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(9, 11, 16, 26) | ||
R | ||
(15, 28, 17, 34) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(5, 13, 27, 28) | 16 | (21, 29, 43, 44) |
(1, 8, 31, 17) | 8 | (9, 16, 39, 25) |
(3, 25, 32, 41) | 20 | (23, 45, 4, 13) |
(10, 14, 29, 38) | 21 | (31, 35, 2, 11) |
(4, 20, 35, 7) | 33 | (37, 5, 20, 40) |
(2, 23, 45, 37) | 44 | (46, 19, 41, 33) |
(15, 34, 39, \({\infty }_{1}\)) | 17 | (32, 3, 8, \({\infty }_{1}\)) |
(19, 46, 44, \({\infty }_{2}\)) | 3 | (22, 1, 47, \({\infty }_{2}\)) |
(21, 22, 33, \({\infty }_{3}\)) | 5 | (26, 27 , 38, \({\infty }_{3}\)) |
(40, 43, 47, \({\infty }_{4}\)) | 15 | (7, 10, 14, \({\infty }_{4}\)) |
(3) (4, 2)-cycle frame of type \(8^{7}4^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 18, 27, 40) | ||
R | ||
(20, 50, 31, 53) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(17, 37, 3, 16) | 51 | (12, 32, 54, 11) |
(29, 5, 41, 53) | 40 | (13, 45, 25, 37) |
(50, 20, 47, 9) | 53 | (47, 17, 44, 6) |
(15, 23, 25, 19) | 36 | (51, 3, 5, 55) |
(36, 33, 51, 52) | 38 | (18, 15, 33, 34) |
(48, 31, 34, 38) | 48 | (40, 23, 26, 30) |
(45, 12, 4, 54) | 54 | (43, 10, 2, 52) |
(43, 6, 1, 26) | 3 | (46, 9, 4, 29) |
(2, 46, 44, \({\infty }_{1}\)) | 34 | (36, 24, 22, \({\infty }_{1}\)) |
(8, 24, 39, \({\infty }_{2}\)) | 33 | (41, 1, 16, \({\infty }_{2}\)) |
(10, 55, 30, \({\infty }_{3}\)) | 9 | (19, 8, 39, \({\infty }_{3}\)) |
(11, 22, 32, \({\infty }_{4}\)) | 16 | (27, 38, 48, \({\infty }_{4}\)) |
(4) (4, 2)-cycle frame of type \(8^{7}8^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 18, 27, 40) | ||
(20, 50, 31, 53) | ||
R | ||
(11, 16, 22, 37) | ||
(19, 29, 54, 44) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(1, 2, 3, 23) | 3 | (4, 5, 6, 26) |
(4, 6, 9, 12) | 41 | (45, 47, 50, 53) |
(8, 16, 10, 19) | 22 | (30, 38, 32, 41) |
(11, 15, 26, 38) | 54 | (9, 13, 24, 36) |
(5, 44, 25, \({\infty }_{1}\)) | 15 | (20, 3, 40, \({\infty }_{1}\)) |
(17, 41, 54, \({\infty }_{2}\)) | 10 | (27, 51, 8, \({\infty }_{2}\)) |
(22, 46, 34, \({\infty }_{3}\)) | 12 | (34, 2, 46, \({\infty }_{3}\)) |
(24, 47, 30, \({\infty }_{4}\)) | 1 | (25, 48, 31, \({\infty }_{4}\)) |
(29, 45, 43, \({\infty }_{5}\)) | 23 | (52, 12, 10, \({\infty }_{5}\)) |
(32, 52, 48, \({\infty }_{6}\)) | 47 | (23, 43, 39, \({\infty }_{6}\)) |
(33, 51, 36, \({\infty }_{7}\)) | 38 | (15, 33, 18, \({\infty }_{7}\)) |
(37, 55, 39, \({\infty }_{8}\)) | 18 | (55, 17, 1, \({\infty }_{8}\)) |
(5) (4, 2)-cycle frame of type \(8^{8}12^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(11, 17, 26, 36) | ||
(14, 21, 35, 52) | ||
(22, 37, 63, 44) | ||
R | ||
(5, 10, 15, 28) | ||
(13, 19, 30, 20) | ||
(18, 31, 45, 60) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(4 , 6, 9, 13) | 53 | (57, 59, 62, 2) |
(1, 2, 3, 42) | 51 | (52, 53, 54, 29) |
(5, 34, 53, \({\infty }_{1}\)) | 36 | (41, 6, 25, \({\infty }_{1}\)) |
(7, 28, 46, \({\infty }_{2}\)) | 35 | (42, 63, 17, \({\infty }_{2}\)) |
(10, 45, 43, \({\infty }_{3}\)) | 55 | (1, 36, 34, \({\infty }_{3}\)) |
(15, 51, 47, \({\infty }_{4}\)) | 63 | (14, 50, 46, \({\infty }_{4}\)) |
(20, 54, 57, \({\infty }_{5}\)) | 1 | (21, 55, 58, \({\infty }_{5}\)) |
(27, 60, 30, \({\infty }_{6}\)) | 41 | (4, 37, 7, \({\infty }_{6}\)) |
(33, 50, 62, \({\infty }_{7}\)) | 11 | (44, 61, 9, \({\infty }_{7}\)) |
(12, 23, 41, \({\infty }_{8}\)) | 10 | (22, 33, 51, \({\infty }_{8}\)) |
(18, 55, 19, \({\infty }_{9}\)) | 20 | (38, 11, 39, \({\infty }_{9}\)) |
(25, 58, 31, \({\infty }_{10}\)) | 18 | (43, 12, 49, \({\infty }_{10}\)) |
(29, 49, 61, \({\infty }_{11}\)) | 38 | (3, 23, 35, \({\infty }_{11}\)) |
(38, 59, 39, \({\infty }_{12}\)) | 52 | (26, 47, 27, \({\infty }_{12}\)) |
(6) (4, 2)-cycle frame of type \(8^{9}8^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 23, 38, 28) | ||
(20, 34, 53, 67) | ||
R | ||
(25, 55, 26, 56) | ||
(19, 40, 21, 50) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(4, 6, 10, 48) | 1 | (5, 7, 11, 49) |
(1, 2, 35, 3) | 12 | (13, 14, 47, 15) |
(12, 29, 16, 60) | 49 | (61, 6, 65, 37) |
(15, 22, 30, 52) | 44 | (59, 66, 2, 24) |
(21, 33, 46, 68) | 55 | (4, 16, 29, 51) |
(26, 37, 69, 65) | 6 | (32, 43, 3, 71) |
(11, 17, 25, 59) | 35 | (46, 52, 60, 22) |
(8, 14, 19, 24) | 34 | (42, 48, 53, 58) |
(5, 51, 71, \({\infty }_{1}\)) | 65 | (70, 44, 64, \({\infty }_{1}\)) |
(7, 44, 41, \({\infty }_{2}\)) | 66 | (1, 38, 35, \({\infty }_{2}\)) |
(31, 43, 66, \({\infty }_{3}\)) | 26 | (57, 69, 20, \({\infty }_{3}\)) |
(32, 58, 57, \({\infty }_{4}\)) | 48 | (8, 34, 33, \({\infty }_{4}\)) |
(39, 56, 40, \({\infty }_{5}\)) | 28 | (67, 12, 68, \({\infty }_{5}\)) |
(42, 62, 55, \({\infty }_{6}\)) | 40 | (10, 30, 23, \({\infty }_{6}\)) |
(47, 70, 49, \({\infty }_{7}\)) | 64 | (39, 62, 41, \({\infty }_{7}\)) |
(50, 61, 64, \({\infty }_{8}\)) | 39 | (17, 28, 31, \({\infty }_{8}\)) |
(7) (4, 2)-cycle frame of type \(8^{9}12^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 23, 38, 28) | ||
(20, 34, 53, 67) | ||
(25, 55, 26, 56) | ||
R | ||
(19, 40, 21, 50) | ||
(32, 70, 37, 71) | ||
(14, 31, 44, 61) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(8, 14, 19, 24) | 43 | (51, 57, 62, 67) |
(11, 17, 29, 22) | 17 | (28, 34, 46, 39) |
(4, 6, 10, 32) | 20 | (24, 26, 30, 52) |
(1, 2, 3, 5) | 3 | (4, 5, 6, 8) |
(7, 57, 60, \({\infty }_{1}\)) | 22 | (29, 7, 10, \({\infty }_{1}\)) |
(12, 64, 48, \({\infty }_{2}\)) | 41 | (53, 33, 17, \({\infty }_{2}\)) |
(16, 62, 51, \({\infty }_{3}\)) | 32 | (48, 22, 11, \({\infty }_{3}\)) |
(30, 65, 52, \({\infty }_{4}\)) | 62 | (20, 55, 42, \({\infty }_{4}\)) |
(40, 68, 44, \({\infty }_{5}\)) | 16 | (56, 12, 60, \({\infty }_{5}\)) |
(50, 58, 61, \({\infty }_{6}\)) | 8 | (58, 66, 69, \({\infty }_{6}\)) |
(15, 35, 42, \({\infty }_{7}\)) | 53 | (68, 16, 23, \({\infty }_{7}\)) |
(21, 33, 59, \({\infty }_{8}\)) | 26 | (47, 59, 13, \({\infty }_{8}\)) |
(31, 66, 43, \({\infty }_{9}\)) | 44 | (3, 38, 15, \({\infty }_{9}\)) |
(37, 69, 46, \({\infty }_{10}\)) | 28 | (65, 25, 2, \({\infty }_{10}\)) |
(39, 71, 47, \({\infty }_{11}\)) | 2 | (41, 1, 49, \({\infty }_{11}\)) |
(41, 49, 70, \({\infty }_{12}\)) | 66 | (35, 43, 64, \({\infty }_{12}\)) |
(8) (4, 2)-cycle frame of type \(8^{13}8^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(20, 43, 70, 45) | ||
(48, 99, 53, 102) | ||
R | ||
(25, 54, 87, 56) | ||
(14, 24, 35, 49) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(41, 83, 24, 82) | 58 | (99, 37, 82, 36) |
(18, 30, 46, 63) | 49 | (67, 79, 95, 8) |
(37, 67, 101, 69) | 66 | (103,29, 63, 31) |
(17, 50, 40, 74) | 81 | (98, 27, 17, 51) |
(8, 14, 61, 72) | 20 | (28, 34, 81, 92) |
(33, 62, 35, 89) | 72 | (1, 30, 3, 57) |
(36, 64, 79, 103) | 22 | (58, 86,101, 21) |
(56, 93, 57, 100) | 27 | (83, 16, 84, 23) |
(10, 19, 25, 3) | 90 | (100, 5, 11, 20) |
(7, 11, 16, 21) | 64 | (71, 75, 80, 85) |
(28, 47, 29, 49) | 12 | (40, 59, 41, 61) |
(42, 86, 44, 97) | 11 | (53, 97, 55, 4) |
(31, 51, 32, 54) | 42 | (73, 93, 74, 96) |
(1, 2, 3, 5) | 41 | (42, 43, 44, 46) |
(4, 6, 9, 12) | 3 | (7, 9, 12, 15) |
(58, 88, 60, 96) | 6 | (64, 94, 66, 102) |
(15, 84, 23, \({\infty }_{1}\)) | 53 | (68, 33, 76, \({\infty }_{1}\)) |
(22, 85, 68, \({\infty }_{2}\)) | 38 | (60, 19, 2, \({\infty }_{2}\)) |
(27, 75, 90, \({\infty }_{3}\)) | 61 | (88, 32, 47, \({\infty }_{3}\)) |
(38, 76, 98, \({\infty }_{4}\)) | 76 | (10, 48, 70, \({\infty }_{4}\)) |
(55, 95, 77, \({\infty }_{5}\)) | 99 | (50, 90, 72, \({\infty }_{5}\)) |
(59, 80, 92, \({\infty }_{6}\)) | 30 | (89, 6, 18, \({\infty }_{6}\)) |
(66, 73, 81, \({\infty }_{7}\)) | 100 | (62, 69, 77, \({\infty }_{7}\)) |
(71, 87, 94, \({\infty }_{8}\)) | 55 | (22, 38, 45, \({\infty }_{8}\)) |
(9) (4, 2)-cycle frame of type \(12^{7}8^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 18, 27, 40) | ||
(5, 8, 10, 23) | ||
R | ||
(25, 48, 26, 59) | ||
(31, 58, 33, 64) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(12, 57, 37, 29) | 43 | (55, 16, 80, 72) |
(82, 30, 69, 9) | 52 | (50, 82, 37, 61) |
(67, 52, 4, 71) | 8 | (75, 60, 12, 79) |
(11, 55, 26, 76) | 76 | (3, 47, 18, 68) |
(2, 39, 17, 33) | 34 | (36, 73, 51, 67) |
(43, 53, 15, 31) | 75 | (34, 44, 6, 22) |
(66, 34, 45, 46) | 20 | (2, 54, 65, 66) |
(36, 74, 48, 72) | 4 | (40, 78, 52, 76) |
(80, 22, 62, 19) | 19 | (15, 41, 81, 38) |
(6, 78, 60, 65) | 51 | (57, 45, 27, 32) |
(1, 79, 3, \({\infty }_{1}\)) | 16 | (17, 11, 19, \({\infty }_{1}\)) |
(16, 59, 58, \({\infty }_{2}\)) | 55 | (71, 30, 29, \({\infty }_{2}\)) |
(20, 24, 61, \({\infty }_{3}\)) | 69 | (5, 9, 46, \({\infty }_{3}\)) |
(25, 54, 44, \({\infty }_{4}\)) | 83 | (24, 53, 43, \({\infty }_{4}\)) |
(41, 47, 50, \({\infty }_{5}\)) | 47 | (4, 10, 13, \({\infty }_{5}\)) |
(64, 73, 75, \({\infty }_{6}\)) | 10 | (74, 83, 1, \({\infty }_{6}\)) |
(32, 51, 81, \({\infty }_{7}\)) | 7 2 | (20, 39, 69, \({\infty }_{7}\)) |
(38, 68, 83, \({\infty }_{8}\)) | 24 | (62, 8, 23, \({\infty }_{8}\)) |
(10) (4, 2)-cycle frame of type \(12^{7}12^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 18, 27, 40) | ||
(5, 8, 10, 23) | ||
(25, 48, 26, 59) | ||
R | ||
(31, 58, 33, 64) | ||
(19, 29, 46, 72) | ||
(20, 30, 53, 75) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(9, 11, 43, 82) | 25 | (34, 36, 68, 23) |
(15, 24, 30, 67) | 72 | (3, 12, 18, 55) |
(22, 33, 51, 34) | 32 | (54, 65, 83, 66) |
(29, 44, 39, 79) | 18 | (47, 62, 57, 13) |
(4, 12, 31, 47) | 13 | (17, 25, 44, 60) |
(1, 2, 3, 61) | 8 | (9, 10, 11, 69) |
(6, 65, 50, \({\infty }_{1}\)) | 2 | (8, 67, 52, \({\infty }_{1}\)) |
(16, 81, 57, \({\infty }_{2}\)) | 29 | (45, 26, 2, \({\infty }_{2}\)) |
(19, 55, 71, \({\infty }_{3}\)) | 19 | (38, 74, 6, \({\infty }_{3}\)) |
(20, 66, 62, \({\infty }_{4}\)) | 59 | (79, 41, 37, \({\infty }_{4}\)) |
(32, 69, 73, \({\infty }_{5}\)) | 16 | (48, 1, 5, \({\infty }_{5}\)) |
(41, 80, 68, \({\infty }_{6}\)) | 75 | (32, 71, 59, \({\infty }_{6}\)) |
(52, 72, 64, \({\infty }_{7}\)) | 36 | (4, 24, 16, \({\infty }_{7}\)) |
(58, 78, 75, \({\infty }_{8}\)) | 3 | (61, 81, 78, \({\infty }_{8}\)) |
(17, 60, 54, \({\infty }_{9}\)) | 22 | (39, 82, 76, \({\infty }_{9}\)) |
(36, 76, 46, \({\infty }_{10}\)) | 4 | (40, 80, 50, \({\infty }_{10}\)) |
(37, 83, 53, \({\infty }_{11}\)) | 74 | (27, 73, 43, \({\infty }_{11}\)) |
(38, 74, 45, \({\infty }_{12}\)) | 61 | (15, 51, 22, \({\infty }_{12}\)) |
(11) (4, 2)-cycle frame of type \(12^{7}16^1\)
1st resolution: | 2nd resolution: | |
|---|---|---|
C | ||
(13, 18, 27, 40) | ||
(19, 29, 46, 72) | ||
(20, 30, 53, 75) | ||
(25, 48, 26, 59) | ||
R | ||
(31, 58, 33, 64) | ||
(34, 79, 36, 81) | ||
(24, 43, 73, 54) | ||
(17, 32, 50, 67) | ||
Starter | Adder | Starter + adder |
|---|---|---|
(1, 2, 3, 5) | 50 | (51, 52, 53, 55) |
(4, 6, 9, 12) | 53 | (57, 59, 62, 65) |
(8, 16, 22, \({\infty }_{1}\)) | 3 | (11, 19, 25, \({\infty }_{1}\)) |
(10, 58, 38, \({\infty }_{2}\)) | 73 | (83, 47, 27, \({\infty }_{2}\)) |
(17, 60, 76, \({\infty }_{3}\)) | 20 | (37, 80, 12, \({\infty }_{3}\)) |
(23, 67, 78, \({\infty }_{4}\)) | 46 | (69, 29, 40, \({\infty }_{4}\)) |
(33, 73, 62, \({\infty }_{5}\)) | 52 | (1, 41, 30, \({\infty }_{5}\)) |
(34, 66, 61, \({\infty }_{6}\)) | 5 | (39, 71, 66, \({\infty }_{6}\)) |
(41, 79, 47, \({\infty }_{7}\)) | 81 | (38, 76, 44, \({\infty }_{7}\)) |
(64, 82, 69, \({\infty }_{8}\)) | 18 | (82, 16, 3, \({\infty }_{8}\)) |
(11, 15, 24, \({\infty }_{9}\)) | 78 | (5, 9, 18, \({\infty }_{9}\)) |
(31, 37, 52, \({\infty }_{10}\)) | 55 | (2, 8, 23, \({\infty }_{10}\)) |
(32, 44, 68, \({\infty }_{11}\)) | 62 | (10, 22, 46, \({\infty }_{11}\)) |
(36, 74, 50, \({\infty }_{12}\)) | 25 | (61, 15, 75, \({\infty }_{12}\)) |
(39, 51, 71, \({\infty }_{13}\)) | 39 | (78, 6, 26, \({\infty }_{13}\)) |
(43, 80, 55, \({\infty }_{14}\)) | 17 | (60, 13, 72, \({\infty }_{14}\)) |
(45, 81, 65, \({\infty }_{15}\)) | 23 | (68, 20, 4, \({\infty }_{15}\)) |
(54, 83, 57, \({\infty }_{16}\)) | 75 | (45, 74, 48, \({\infty }_{16}\)) |
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Wang, J., Xie, L. Doubly Resolvable 4-Cycle Systems of \(2K_v\). Graphs and Combinatorics 34, 313–337 (2018). https://doi.org/10.1007/s00373-018-1875-y
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DOI: https://doi.org/10.1007/s00373-018-1875-y