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. ChangP. J. Dukes and T. Feng, Leaves for packings with block size four, Austral. J. Combin., 80 (2021), 281-304. 

[7]

Y. ChangG. 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. ChangG. Lo FaroA. 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. FrancetićP. 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. GaoY. 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. GaoY. 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. GeS. HuE. 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. GeE. 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. HoffmanC. C. LindnerM. 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. HuY. 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. MendelsohnN. 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. WeiG. 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. ChangP. J. Dukes and T. Feng, Leaves for packings with block size four, Austral. J. Combin., 80 (2021), 281-304. 

[7]

Y. ChangG. 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. ChangG. Lo FaroA. 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. FrancetićP. 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. GaoY. 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. GaoY. 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. GeS. HuE. 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. GeE. 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. HoffmanC. C. LindnerM. 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. HuY. 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. MendelsohnN. 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. WeiG. 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. 

Table 1.  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 $
Table 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 $
Table 3.  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 $
Table 4.  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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