# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022040
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## Kite-group divisible packings and coverings with any minimum leave and minimum excess

 1 Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, China 2 School of Intelligence Policing, China People's Police University, Langfang 065000, China

*Corresponding author: Yanxun Chang

Received  February 2022 Revised  April 2022 Early access June 2022

Fund Project: This work is supported by NSFC under Grant 11971053 (Y. Chang), NSFC under Grant 11901210 (L. Wang)

Following Hu, Chang and Feng's work [Graphs Combin., 2016], we further subdivide the possible types of minimum leaves and minimum excesses for maximum group divisible packings and minimum group divisible coverings with kites. We show that a maximum group divisible packing and a minimum group divisible covering with kites for any given type of minimum leave and minimum excess exist, respectively.

Citation: Yuxing Yang, Yanxun Chang, Lidong Wang. Kite-group divisible packings and coverings with any minimum leave and minimum excess. Advances in Mathematics of Communications, doi: 10.3934/amc.2022040
##### References:
 [1] R. J. R. Abel, F.E. Bennett and M. Greig, PBD-closure, CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, (2007), 247-255. [2] A. Assaf, Modified group divisible designs, Ars Combin., 29 (1990), 13-20. [3] E. J. Billington and C. C. Lindner, Maximum packings of uniform group divisible triple systems, J. Combin. Des., 4 (1996), 397-404.  doi: 10.1002/(SICI)1520-6610(1996)4:6<397::AID-JCD2>3.0.CO;2-A. [4] J.-C. Bermond and J. Schönheim, $G$-decomposition of $K_n$, where $G$ has four vertices or less, Discrete Math., 19 (1977), 113-120.  doi: 10.1016/0012-365X(77)90027-9. [5] A. E. Brouwer, Optimal packings of $K_{4}$'s into a $K_{n}$, J. Combin. Theory Ser. A, 26 (1979), 278-297.  doi: 10.1016/0097-3165(79)90105-5. [6] Y. Chang, P. J. Dukes and T. Feng, Leaves for packings with block size four, Austral. J. Combin., 80 (2021), 281-304. [7] Y. Chang, G. Lo Faro and A. Tripodi, Tight blocking sets in some maximum packings of $\lambda K_n$, Discrete Math., 308 (2008), 427-438.  doi: 10.1016/j.disc.2006.11.060. [8] Y. Chang, G. Lo Faro, A. Tripodi and J. Zhou, TBSs in some minimum coverings, Discrete Math., 313 (2013), 278-285.  doi: 10.1016/j.disc.2012.10.004. [9] R. Dutta and G. N. Rouskas, Traffic grooming in WDM networks: Past and future, IEEE Network, 16 (2002), 46-56. [10] M. K. Fort Jr. and G. A. Hedlund, Minimum coverings of pairs by triples, Pacific J. Math., 8 (1958), 709-719.  doi: 10.2140/pjm.1958.8.709. [11] N. Francetić, P. Danziger and E. Mendelsohn, Group divisible covering designs with block size 4: a type of covering array with row limit, J. Combin. Des., 21 (2013), 311-341.  doi: 10.1002/jcd.21324. [12] Y. Gao, Y. Chang and T. Feng, Group divisible $(K_4-e)$-packings with any minimum leave, J. Combin. Des., 26 (2018), 315-343.  doi: 10.1002/jcd.21600. [13] Y. Gao, Y. Chang and T. Feng, Simple minimum $(K_4-e)$-coverings of complete multipartite graphs, Acta Math. Sin., 35 (2019), 632-648.  doi: 10.1007/s10114-019-8005-5. [14] G. Ge, S. Hu, E. Koloto$\breve{g}$lu and H. Wei, A complete solution to spectrum problem for five-vertex graphs with application to traffic grooming in optical networks, J. Combin. Des., 23 (2015), 233-273.  doi: 10.1002/jcd.21405. [15] G. Ge, E. Koloto$\breve{g}$lu and H. Wei, Optimal groomings with grooming ratios six and seven, J. Combin. Des., 23 (2015), 400-415.  doi: 10.1002/jcd.21428. [16] G. Haggard, On the function $N(3, 2, \lambda, v)$, Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Florida Atlantic Univ., Boca Raton, Fla., 6 (1972), 243-250. [17] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., 11 (1975) 255-369. doi: 10.1016/0012-365X(75)90040-0. [18] K. Heinrich and J. Yin, On group divisible covering designs, Discrete Math., 202 (1999), 101-112.  doi: 10.1016/S0012-365X(98)00362-8. [19] D. G. Hoffman, C. C. Lindner, M. J. Sharry and A. P. Street, Maximum packings of $K_n$ with copies of $(K_4-e)$, Aequationes Math., 51 (1996), 247-269.  doi: 10.1007/BF01833281. [20] D. G. Hoffman and K. Kirkpatrick, $G$-designs of order $n$ and index $\lambda$ when $G$ has $5$ vertices or less, Austral. J. Combin., 18 (1998), 13-37. [21] X. Hu, Y. Chang and T. Feng, Group divisible packings and coverings with any minimum leave and minimum excess, Graphs Combin., 32 (2016), 1423-1446.  doi: 10.1007/s00373-015-1644-0. [22] C. C. Lindner and A. P. Street, Simple minimum coverings of $K_n$ with copies of $(K_4-e)$, Aequationes Math., 52 (1996), 284-301.  doi: 10.1007/BF01818345. [23] E. Mendelsohn, N. Shalaby and H. Shen, Nuclear designs, Ars Combin., 32 (1991), 225-238. [24] W. H. Mills, On the covering of pairs by quadruples. I, J. Combin. Theory Ser. A, 13 (1972), 55-78.  doi: 10.1016/0097-3165(72)90008-8. [25] W. H. Mills, On the covering of pairs by quadruples. II, J. Combin. Theory Ser. A, 15 (1973), 138-166.  doi: 10.1016/S0097-3165(73)80003-2. [26] J. Spencer, Maximal consistent families for triples, J. Combin. Theory Ser. A, 5 (1968), 1-8.  doi: 10.1016/S0021-9800(68)80023-7. [27] R. G. Stanton and M. J. Rogers, Packings and coverings by triples, Ars Combin., 13 (1982), 61-69. [28] D. R. Stinson, Combinatorial Designs: Constructions and Analysis, Springer-Verlag, New York, 2004. [29] H. Wang and Y. Chang, Kite-group divisible designs of type $g^{t}u^{1}$, Graphs Combin., 22 (2006), 545-571.  doi: 10.1007/s00373-006-0681-0. [30] H. Wang and Y. Chang, $(K_3+e, \lambda)$-group divisible designs of type $g^{t}u^{1}$, Ars Combin., 89 (2008), 63-88. [31] J. Wang, Incomplete group divisible designs with block size four, J. Combin. Des., 11 (2003), 442-455.  doi: 10.1002/jcd.10055. [32] H. Wei, G. Ge and C. J. Colbourn, Group divisible covering designs with block size four, J. Combin. Des., 26 (2018), 101-118.  doi: 10.1002/jcd.21596. [33] R. M. Wilson, Constructions and uses of pairwise balanced designs, Math. Centre Tracts, Math. Centrum, Amsterdam, 55 (1974), 18-41. [34] J. Yin, Packing designs with equal-sized holes, J. Statist. Plann. Inference, 94 (2001), 393-403.  doi: 10.1016/S0378-3758(00)00269-X. [35] J. Yin and J. Wang, $(3, \lambda$)-group divisible covering designs, Austral. J. Combin., 15 (1997), 61-70.

show all references

##### References:
 [1] R. J. R. Abel, F.E. Bennett and M. Greig, PBD-closure, CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, (2007), 247-255. [2] A. Assaf, Modified group divisible designs, Ars Combin., 29 (1990), 13-20. [3] E. J. Billington and C. C. Lindner, Maximum packings of uniform group divisible triple systems, J. Combin. Des., 4 (1996), 397-404.  doi: 10.1002/(SICI)1520-6610(1996)4:6<397::AID-JCD2>3.0.CO;2-A. [4] J.-C. Bermond and J. Schönheim, $G$-decomposition of $K_n$, where $G$ has four vertices or less, Discrete Math., 19 (1977), 113-120.  doi: 10.1016/0012-365X(77)90027-9. [5] A. E. Brouwer, Optimal packings of $K_{4}$'s into a $K_{n}$, J. Combin. Theory Ser. A, 26 (1979), 278-297.  doi: 10.1016/0097-3165(79)90105-5. [6] Y. Chang, P. J. Dukes and T. Feng, Leaves for packings with block size four, Austral. J. Combin., 80 (2021), 281-304. [7] Y. Chang, G. Lo Faro and A. Tripodi, Tight blocking sets in some maximum packings of $\lambda K_n$, Discrete Math., 308 (2008), 427-438.  doi: 10.1016/j.disc.2006.11.060. [8] Y. Chang, G. Lo Faro, A. Tripodi and J. Zhou, TBSs in some minimum coverings, Discrete Math., 313 (2013), 278-285.  doi: 10.1016/j.disc.2012.10.004. [9] R. Dutta and G. N. Rouskas, Traffic grooming in WDM networks: Past and future, IEEE Network, 16 (2002), 46-56. [10] M. K. Fort Jr. and G. A. Hedlund, Minimum coverings of pairs by triples, Pacific J. Math., 8 (1958), 709-719.  doi: 10.2140/pjm.1958.8.709. [11] N. Francetić, P. Danziger and E. Mendelsohn, Group divisible covering designs with block size 4: a type of covering array with row limit, J. Combin. Des., 21 (2013), 311-341.  doi: 10.1002/jcd.21324. [12] Y. Gao, Y. Chang and T. Feng, Group divisible $(K_4-e)$-packings with any minimum leave, J. Combin. Des., 26 (2018), 315-343.  doi: 10.1002/jcd.21600. [13] Y. Gao, Y. Chang and T. Feng, Simple minimum $(K_4-e)$-coverings of complete multipartite graphs, Acta Math. Sin., 35 (2019), 632-648.  doi: 10.1007/s10114-019-8005-5. [14] G. Ge, S. Hu, E. Koloto$\breve{g}$lu and H. Wei, A complete solution to spectrum problem for five-vertex graphs with application to traffic grooming in optical networks, J. Combin. Des., 23 (2015), 233-273.  doi: 10.1002/jcd.21405. [15] G. Ge, E. Koloto$\breve{g}$lu and H. Wei, Optimal groomings with grooming ratios six and seven, J. Combin. Des., 23 (2015), 400-415.  doi: 10.1002/jcd.21428. [16] G. Haggard, On the function $N(3, 2, \lambda, v)$, Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Florida Atlantic Univ., Boca Raton, Fla., 6 (1972), 243-250. [17] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., 11 (1975) 255-369. doi: 10.1016/0012-365X(75)90040-0. [18] K. Heinrich and J. Yin, On group divisible covering designs, Discrete Math., 202 (1999), 101-112.  doi: 10.1016/S0012-365X(98)00362-8. [19] D. G. Hoffman, C. C. Lindner, M. J. Sharry and A. P. Street, Maximum packings of $K_n$ with copies of $(K_4-e)$, Aequationes Math., 51 (1996), 247-269.  doi: 10.1007/BF01833281. [20] D. G. Hoffman and K. Kirkpatrick, $G$-designs of order $n$ and index $\lambda$ when $G$ has $5$ vertices or less, Austral. J. Combin., 18 (1998), 13-37. [21] X. Hu, Y. Chang and T. Feng, Group divisible packings and coverings with any minimum leave and minimum excess, Graphs Combin., 32 (2016), 1423-1446.  doi: 10.1007/s00373-015-1644-0. [22] C. C. Lindner and A. P. Street, Simple minimum coverings of $K_n$ with copies of $(K_4-e)$, Aequationes Math., 52 (1996), 284-301.  doi: 10.1007/BF01818345. [23] E. Mendelsohn, N. Shalaby and H. Shen, Nuclear designs, Ars Combin., 32 (1991), 225-238. [24] W. H. Mills, On the covering of pairs by quadruples. I, J. Combin. Theory Ser. A, 13 (1972), 55-78.  doi: 10.1016/0097-3165(72)90008-8. [25] W. H. Mills, On the covering of pairs by quadruples. II, J. Combin. Theory Ser. A, 15 (1973), 138-166.  doi: 10.1016/S0097-3165(73)80003-2. [26] J. Spencer, Maximal consistent families for triples, J. Combin. Theory Ser. A, 5 (1968), 1-8.  doi: 10.1016/S0021-9800(68)80023-7. [27] R. G. Stanton and M. J. Rogers, Packings and coverings by triples, Ars Combin., 13 (1982), 61-69. [28] D. R. Stinson, Combinatorial Designs: Constructions and Analysis, Springer-Verlag, New York, 2004. [29] H. Wang and Y. Chang, Kite-group divisible designs of type $g^{t}u^{1}$, Graphs Combin., 22 (2006), 545-571.  doi: 10.1007/s00373-006-0681-0. [30] H. Wang and Y. Chang, $(K_3+e, \lambda)$-group divisible designs of type $g^{t}u^{1}$, Ars Combin., 89 (2008), 63-88. [31] J. Wang, Incomplete group divisible designs with block size four, J. Combin. Des., 11 (2003), 442-455.  doi: 10.1002/jcd.10055. [32] H. Wei, G. Ge and C. J. Colbourn, Group divisible covering designs with block size four, J. Combin. Des., 26 (2018), 101-118.  doi: 10.1002/jcd.21596. [33] R. M. Wilson, Constructions and uses of pairwise balanced designs, Math. Centre Tracts, Math. Centrum, Amsterdam, 55 (1974), 18-41. [34] J. Yin, Packing designs with equal-sized holes, J. Statist. Plann. Inference, 94 (2001), 393-403.  doi: 10.1016/S0378-3758(00)00269-X. [35] J. Yin and J. Wang, $(3, \lambda$)-group divisible covering designs, Austral. J. Combin., 15 (1997), 61-70.
Realizable leaves among $(K_3+e,\lambda)$-MGDPs of type $g^n$, $(g,n,\lambda)\neq(1,4,1)$
 $g\ ({\rm mod}\ 2)$ $n\ ({\rm mod}\ 8)$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $\lambda=1$ $\emptyset$ $E_1$ $E_{31}$, $E_{32}$, $E_{33}$, $E_{34}$, $E_{35}$ $E_{21}$, $E_{22}$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\lambda$ $\pmod{4}$ 1 $\emptyset$ $E_1$ $E_3$ $E_2$ $\lambda > 1$ 2 $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $\emptyset$ $E_3$ $E_1$ $E_2$
 $g\ ({\rm mod}\ 2)$ $n\ ({\rm mod}\ 8)$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $\lambda=1$ $\emptyset$ $E_1$ $E_{31}$, $E_{32}$, $E_{33}$, $E_{34}$, $E_{35}$ $E_{21}$, $E_{22}$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\lambda$ $\pmod{4}$ 1 $\emptyset$ $E_1$ $E_3$ $E_2$ $\lambda > 1$ 2 $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $\emptyset$ $E_3$ $E_1$ $E_2$
Realizable excesses among $(K_3+e,\lambda)$-MGDCs of type $g^n$, $(g,n,\lambda)\neq(1,4,1)$
 $g\ ({\rm mod}\ 2)$ $n\ ({\rm mod}\ 8)$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $\lambda$ 1 $\emptyset$ $E_3$ $E_1$ $E_2$ ($\rm mod$ 4) 2 $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $\emptyset$ $E_1$ $E_3$ $E_2$
 $g\ ({\rm mod}\ 2)$ $n\ ({\rm mod}\ 8)$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $\lambda$ 1 $\emptyset$ $E_3$ $E_1$ $E_2$ ($\rm mod$ 4) 2 $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $\emptyset$ $E_1$ $E_3$ $E_2$
Possible leaves among $(K_3+e,\lambda)$-MGDPs of type $g^n$
 $g\ ({\rm mod}\ 2)$ $n=3$ $n\ ({\rm mod}\ 8)$, $n\geq4$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $\lambda=1$ $E_{31}$, $E_{32}(6,2)$, $E_{32}(6,3)$, $E_{33}(5,2)$, $E_{33}(5,3)$, $\emptyset$ $E_1$ $E_{31}$, $E_{32}$, $E_{33}$, $E_{21}$, $E_{22}$ $E_{34}(4,3)$, $E_{34}(4,2)$, $E_{35}(4,3)$, $E_{35}(4,2)$ $E_{34}$, $E_{35}$ $\lambda$ $\pmod{4} $$\lambda > 1 0 \emptyset \emptyset \emptyset \emptyset \emptyset 1 E_{31} , E_{32}(6,2) , E_{32}(6,3) , E_{33}(5,2) , E_{33}(5,3) , E_{34}(4,3) , E_{34}(4,2) , E_{35}(4,3) , E_{35}(4,2) , \emptyset E_1 E_3 E_2 E_{36}(4,2) , E_{36}(4,3) , E_{37} , E_{38} 2 E_{21}(4,3) , E_{21}(4,2) , E_{22}(3,3) , \emptyset E_2 E_2 \emptyset E_{22}(3,2) , E_{23}(2,2) 3 E_1 \emptyset E_3 E_1 E_2  g\ ({\rm mod}\ 2) n=3 n\ ({\rm mod}\ 8) , n\geq4 0, 1 2, 7 3, 6 4, 5 0 \lambda\geq 1 \emptyset \emptyset \emptyset \emptyset \emptyset 1 \lambda=1 E_{31} , E_{32}(6,2) , E_{32}(6,3) , E_{33}(5,2) , E_{33}(5,3) , \emptyset E_1 E_{31} , E_{32} , E_{33} , E_{21} , E_{22} E_{34}(4,3) , E_{34}(4,2) , E_{35}(4,3) , E_{35}(4,2) E_{34} , E_{35} \lambda \pmod{4}$$ \lambda > 1$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 $E_{31}$, $E_{32}(6,2)$, $E_{32}(6,3)$, $E_{33}(5,2)$, $E_{33}(5,3)$, $E_{34}(4,3)$, $E_{34}(4,2)$, $E_{35}(4,3)$, $E_{35}(4,2)$, $\emptyset$ $E_1$ $E_3$ $E_2$ $E_{36}(4,2)$, $E_{36}(4,3)$, $E_{37}$, $E_{38}$ 2 $E_{21}(4,3)$, $E_{21}(4,2)$, $E_{22}(3,3)$, $\emptyset$ $E_2$ $E_2$ $\emptyset$ $E_{22}(3,2)$, $E_{23}(2,2)$ 3 $E_1$ $\emptyset$ $E_3$ $E_1$ $E_2$
Possible excesses among $(K_3+e,\lambda)$-MGDCs of type $g^n$
 $g\ ({\rm mod}\ 2)$ $n=3$ $n\ ({\rm mod}\ 8)$, $n\geq4$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 1 $E_1$ $\emptyset$ $E_3$ $E_1$ $E_2$ $\lambda$ $\pmod{4}$ 2 $E_{21}(4,3)$, $E_{21}(4,2)$, $E_{22}(3,3)$, $E_{22}(3,2)$, $E_{23}(2,2)$ $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $E_{31}$, $E_{32}(6,2)$, $E_{32}(6,3)$, $E_{33}(5,2)$, $E_{33}(5,3)$, $E_{34}(4,3)$, $E_{34}(4,2)$, $E_{35}(4,3)$, $E_{35}(4,2)$, $E_{36}(4,2)$, $E_{36}(4,3)$, $E_{37}$, $E_{38}$ $\emptyset$ $E_1$ $E_3$ $E_2$
 $g\ ({\rm mod}\ 2)$ $n=3$ $n\ ({\rm mod}\ 8)$, $n\geq4$ 0, 1 2, 7 3, 6 4, 5 0 $\lambda\geq 1$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 0 $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ $\emptyset$ 1 1 $E_1$ $\emptyset$ $E_3$ $E_1$ $E_2$ $\lambda$ $\pmod{4}$ 2 $E_{21}(4,3)$, $E_{21}(4,2)$, $E_{22}(3,3)$, $E_{22}(3,2)$, $E_{23}(2,2)$ $\emptyset$ $E_2$ $E_2$ $\emptyset$ 3 $E_{31}$, $E_{32}(6,2)$, $E_{32}(6,3)$, $E_{33}(5,2)$, $E_{33}(5,3)$, $E_{34}(4,3)$, $E_{34}(4,2)$, $E_{35}(4,3)$, $E_{35}(4,2)$, $E_{36}(4,2)$, $E_{36}(4,3)$, $E_{37}$, $E_{38}$ $\emptyset$ $E_1$ $E_3$ $E_2$
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