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Group Divisible Packings and Coverings with Any Minimum Leave and Minimum Excess

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Abstract

We examine all possible minimum leaves and minimum excesses for maximum group divisible packings and minimum group divisible coverings with triangles or kites, respectively. Necessary and sufficient conditions are established for their existences.

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Authors and Affiliations

  1. Institute of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Xijuan Hu, Yanxun Chang & Tao Feng

Authors
  1. Xijuan Hu
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  2. Yanxun Chang
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  3. Tao Feng
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Corresponding author

Correspondence to Tao Feng.

Additional information

Supported by NSFC under Grants 11271042 and 11471032.

Appendices

Appendix 1: (3, 1)-MGDC of type \(5^4\) from [17]

The required (3, 1)-MGDC of type \(5^4\) is constructed on \(S \cup R\) where \(S=\{a_{ij}:1 \le i \le 3\) and \(1 \le j \le 5\}\) and \(R=\{x_j:1 \le j \le 5\}\) with \(S \cap R=\phi \). The four groups are R and \(\{a_{ij}:1\le j\le 5\}\) for \(1\le i\le 3\). All blocks are listed below.

$$\begin{aligned} \begin{array}{lllll} \{x_{1},a_{12},a_{24}\} &{} \{x_{1},a_{22},a_{34}\} &{} \{x_{1},a_{32},a_{14}\} &{} \{x_{1},a_{13},a_{25}\} &{} \{x_{1},a_{23},a_{35}\}\\ \{x_{1},a_{33},a_{15}\} &{} \{x_{2},a_{11},a_{25}\} &{} \{x_{2},a_{21},a_{35}\} &{} \{x_{2},a_{31},a_{15}\} &{} \{x_{2},a_{13},a_{24}\}\\ \{x_{2},a_{23},a_{34}\} &{} \{x_{2},a_{33},a_{14}\} &{} \{x_{3},a_{11},a_{24}\} &{} \{x_{3},a_{21},a_{34}\} &{} \{x_{3},a_{31},a_{14}\}\\ \{x_{3},a_{12},a_{25}\} &{} \{x_{3},a_{22},a_{35}\} &{} \{x_{3},a_{32},a_{15}\} &{} \{x_{4},a_{14},a_{22}\} &{} \{x_{4},a_{24},a_{32}\}\\ \{x_{4},a_{34},a_{12}\} &{} \{x_{4},a_{14},a_{21}\} &{} \{x_{4},a_{24},a_{31}\} &{} \{x_{4},a_{34},a_{11}\} &{} \{x_{4},a_{15},a_{23}\}\\ \{x_{4},a_{25},a_{33}\} &{} \{x_{4},a_{35},a_{13}\} &{} \{x_{5},a_{15},a_{21}\} &{} \{x_{5},a_{25},a_{31}\} &{} \{x_{5},a_{35},a_{11}\}\\ \{x_{5},a_{15},a_{22}\} &{} \{x_{5},a_{25},a_{32}\} &{} \{x_{5},a_{35},a_{12}\} &{} \{x_{5},a_{14},a_{23}\} &{} \{x_{5},a_{24},a_{33}\}\\ \{x_{5},a_{34},a_{13}\} &{} \{a_{14},a_{25},a_{34}\} &{} \{a_{24},a_{35},a_{14}\} &{} \{a_{34},a_{15},a_{24}\} &{} \{a_{12},a_{21},a_{33}\}\\ \{a_{22},a_{31},a_{13}\} &{} \{a_{32},a_{11},a_{23}\} &{} \{a_{15},a_{25},a_{35}\} &{} \{x_{1},a_{11},a_{21}\} &{} \{x_{1},a_{11},a_{31}\}\\ \{x_{2},a_{12},a_{22}\} &{} \{x_{2},a_{22},a_{32}\} &{} \{x_{3},a_{13},a_{33}\} &{} \{x_{3},a_{23},a_{33}\} &{} \{a_{11},a_{22},a_{33}\}\\ \{a_{13},a_{23},a_{31}\} &{} \{a_{12},a_{21},a_{31}\} &{} \{a_{13},a_{21},a_{32}\} &{} \{a_{12},a_{32},a_{23}\} &{} \end{array} \end{aligned}$$

The excess graph contains edges: \(\{x_{4},a_{14}\}\), \(\{x_{4},a_{24}\}\), \(\{x_{4},a_{34}\}\), \(\{x_{5},a_{15}\}\), \(\{x_{5},a_{25}\}\), \(\{x_{5},a_{35}\}\), \(\{x_{1},a_{11}\}\), \(\{x_{2},a_{22}\}\), \(\{x_{3},a_{33}\}\), \(\{a_{12},a_{21}\}\), \(\{a_{13},a_{31}\}\), \(\{a_{23},a_{32}\}\).

Appendix 2: (3, 1)-IGDC of type \((11,5)^4\) in Lemma 4.5

$$\begin{aligned} \begin{array}{llllll} \{0,2,25\}&{} \{3,4,25\}&{} \{6,7,25\}&{} \{8,10,25\}&{} \{11,12,25\}&{} \{14,15,25\}\\ \{16,18,25\}&{} \{19,20,25\}&{} \{22,23,25\}&{} \{0,1,27\}&{} \{4,10,27\}&{} \{5,18,27\}\\ \{2,8,27\}&{} \{6,9,27\}&{} \{12,13,27\}&{} \{14,16,27\}&{} \{17,20,27\}&{} \{21,22,27\}\\ \{0,5,31\}&{} \{1,10,31\}&{} \{2,16,31\}&{} \{6,12,31\}&{} \{4,17,31\}&{} \{8,21,31\}\\ \{9,14,31\}&{} \{13,18,31\}&{} \{20,22,31\}&{} \{0,6,35\}&{} \{0,9,39\}&{} \{0,10,43\}\\ \{5,10,39\}&{} \{1,16,39\}&{} \{4,22,39\}&{} \{6,8,39\}&{} \{2,13,39\}&{} \{12,17,39\}\\ \{14,20,39\}&{} \{18,21,39\}&{} \{9,10,35\}&{} \{5,16,35\}&{} \{4,13,35\}&{} \{1,22,35\}\\ \{2,8,35\}&{} \{14,17,35\}&{} \{12,18,35\}&{} \{20,21,35\}&{} \{6,13,43\}&{} \{9,16,43\}\\ \{2,17,43\}&{} \{5,22,43\}&{} \{1,8,43\}&{} \{12,14,43\}&{} \{18,20,43\}&{} \{4,21,43\}\\ \{1,4,26\}&{} \{1,4,30\}&{} \{4,5,34\}&{} \{4,7,38\}&{} \{4,9,42\}&{} \{1,3,38\}\\ \{5,12,38\}&{} \{9,20,38\}&{} \{8,11,38\}&{} \{0,13,38\}&{} \{15,16,38\}&{} \{17,19,38\}\\ \{21,23,38\}&{} \{1,2,12\}&{} \{1,7,34\}&{} \{1,11,42\}&{} \{1,6,20\}&{} \{1,14,24\}\\ \{1,15,28\}&{} \{1,18,32\}&{} \{1,19,36\}&{} \{1,23,40\}&{} \{3,20,34\}&{} \{9,12,34\}\\ \{8,13,34\}&{} \{0,15,34\}&{} \{16,23,34\}&{} \{11,17,34\}&{} \{19,21,34\}&{} \{5,20,42\}\\ \{8,15,42\}&{} \{0,23,42\}&{} \{3,12,42\}&{} \{7,21,42\}&{} \{13,16,42\}&{} \{17,19,42\}\\ \{10,12,21\}&{} \{7,12,26\}&{} \{12,15,30\}&{} \{12,15,29\}&{} \{12,19,33\}&{} \{12,22,37\}\\ \{12,23,41\}&{} \{11,20,26\}&{} \{7,20,30\}&{} \{0,3,26\}&{} \{15,21,26\}&{} \{5,8,26\}\\ \{9,19,26\}&{} \{13,16,26\}&{} \{17,23,26\}&{} \{3,5,30\}&{} \{9,11,30\}&{} \{13,19,30\}\\ \{8,23,30\}&{} \{0,17,30\}&{} \{16,21,30\}&{} \{0,3,21\}&{} \{3,8,9\}&{} \{7,8,17\}\\ \{8,14,33\}&{} \{8,18,37\}&{} \{8,19,29\}&{} \{8,22,41\}&{} \{0,7,33\}&{} \{0,11,22\}\\ \{0,14,29\}&{} \{0,19,37\}&{} \{0,18,41\}&{} \{2,20,29\}&{} \{10,20,33\}&{} \{14,20,41\}\\ \{13,15,20\}&{} \{20,23,37\}&{} \{2,7,41\}&{} \{3,6,41\}&{} \{4,15,41\}&{} \{10,11,41\}\\ \{16,19,41\}&{} \{7,10,37\}&{} \{7,16,22\}&{} \{7,18,29\}&{} \{2,15,37\}&{} \{3,14,37\}\\ \{4,6,37\}&{} \{11,16,37\}&{} \{4,11,29\}&{} \{4,14,19\}&{} \{2,4,33\}&{} \{4,18,23\}\\ \{6,16,29\}&{} \{3,16,33\}&{} \{10,16,17\}&{} \{3,10,29\}&{} \{22,23,29\}&{} \{15,22,33\}\\ \{11,18,33\}&{} \{6,23,33\}&{} \{5,7,24\}&{} \{18,19,24\}&{} \{10,19,28\}&{} \{19,22,32\}\\ \{2,5,19\}&{} \{6,19,40\}&{} \{2,3,32\}&{} \{6,17,32\}&{} \{11,21,32\}&{} \{2,21,24\}\\ \{2,9,40\}&{} \{2,11,28\}&{} \{2,23,36\}&{} \{6,21,36\}&{} \{5,6,11\}&{} \{6,9,28\}\\ \{6,15,24\}&{} \{7,9,36\}&{} \{9,15,18\}&{} \{9,22,24\}&{} \{9,23,32\}&{} \{3,17,24\}\\ \{5,23,28\}&{} \{13,14,23\}&{} \{10,23,24\}&{} \{11,13,24\}&{} \{5,14,36\}&{} \{11,14,40\}\\ \{10,11,36\}&{} \{14,21,28\}&{} \{7,14,32\}&{} \{5,7,40\}&{} \{5,15,32\}&{} \{7,13,28\}\\ \{10,15,40\}&{} \{10,13,32\}&{} \{15,17,36\}&{} \{18,21,40\}&{} \{17,18,28\}&{} \{3,18,36\}\\ \{3,22,28\}&{} \{3,13,40\}&{} \{13,22,36\}&{} \{17,22,40\} \end{array} \end{aligned}$$

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Hu, X., Chang, Y. & Feng, T. Group Divisible Packings and Coverings with Any Minimum Leave and Minimum Excess. Graphs and Combinatorics 32, 1423–1446 (2016). https://doi.org/10.1007/s00373-015-1644-0

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