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On Decompositions of Complete 3-Uniform Hypergraphs into a Linear Forest with 4 Edges

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Combinatorics, Graph Theory and Computing (SEICCGTC 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 448))

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Abstract

A 3-uniform linear forest is any hypergraph obtained by starting with a single 3-uniform edge and adding other 3-uniform edges sequentially such that each additional edge intersects with the previous hypergraph at no more than one vertex. There are nine such 3-uniform linear forests with four edges. In this paper we establish necessary and sufficient conditions for a decomposition of a complete 3-uniform hypergraph into isomorphic copies of a linear forest with four edges.

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Acknowledgements

This research is supported by grant number A1659815 from the Division of Mathematical Sciences at the National Science Foundation. This work was done while all but the first author were participants in REU Site: Mathematics Research Experience for Pre-service and for In-service Teachers at Illinois State University.

Author information

Authors and Affiliations

  1. Illinois State University, Normal, IL, USA

    Ryan C. Bunge & Mary Donovan

  2. Colorado State University, Fort Collins, CO, USA

    Erin Dawson

  3. Grayslake Central High School, Grayslake, IL, USA

    Cody Hatzer

  4. Niagara University, Niagara Falls, NY, USA

    Jacquelyn Maass

Authors
  1. Ryan C. Bunge
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  2. Erin Dawson
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  3. Mary Donovan
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  4. Cody Hatzer
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  5. Jacquelyn Maass
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Corresponding author

Correspondence to Ryan C. Bunge .

Editor information

Editors and Affiliations

  1. Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA

    Frederick Hoffman

  2. Mathematics, Kennesaw State University, Kennesaw, GA, USA

    Sarah Holliday

  3. Mathematics, Florida Atlantic University, Boca Raton, FL, USA

    Zvi Rosen

  4. Computer Science and Engineering, University of North Texas, Denton, TX, USA

    Farhad Shahrokhi

  5. Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, USA

    John Wierman

Appendix A

Appendix A

We now give the remaining \(H_k\)-decompositions not explicitly shown in Sect. 2. As with Examples 3 through 5, if \(H_k\) has \(r_k\) vertices, the \(H_k\)-blocks listed in sets \(B_k\) and \(B'_k\) are displayed as \([v_1,v_2, \ldots ,v_{r_k}]\) rather than as \(H_k[v_1,v_2, \ldots ,v_{r_k}]\).

Example 6

Let \(V \Big (K^{(3)}_{12}\Bigr ) = \mathbb {Z}_{11} \cup \{f_1\}\) and let

$$\begin{aligned} B_1 = \bigl \{&[10,3,0,1,2,6,7,8,f_1],\, [10,7,0,1,6,2,3,5,f_1],\, [9,3,0,2,4,5,7,10,f_1], \\ & [10,6,0,1,3,2,7,9,4],\, [8,4,0,3,6,7,f_1,5,10]\bigr \}, \\ B_2=\bigl \{&[f_1,0,10,1,9,2,8,3,5,7],\, [0,1,4,3,6,2,f_1,7,8,9],\, [0,1,3,2,5,4,f_1,6,8,9], \\ & [0,1,5,2,6,4,7,3,8,9],\, [1,3,9,4,5,7,0,2,6,8]\bigr \}, \\ B_3 = \bigl \{&[1,2,0,3,6,7,10,4,8,9],\, [1,8,0,2,7,5,3,9,10,f_1],\, [2,5,0,1,6,7,9,3,4,8], \\ & [3,7,0,4,f_1,5,10,6,8,9],\, [2,6,0,3,f_1,7,9,5,8,10]\bigr \}, \\ B_4=\bigl \{&[0,1,2,3,5,6,8,10,f_1,4],\, [0,1,4,f_1,8,5,7,10,2,6],\, [0,1,6,8,3,10,2,5,f_1,7], \\ & [0,1,5,6,3,2,7,9,f_1,10],\, [0,1,7,8,4,5,10,3,f_1,6]\bigr \}, \\ B_5 = \bigl \{&[0,1,f_1,4,10,3,7,9,5,6,8],\, [0,2,f_1,1,5,3,4,9,6,8,10],\, [1,5,0,2,6,3,4,10,7,8,9], \\ & [2,8,0,3,7,1,4,10,5,6,9],\, [3,10,2,4,5,1,6,9,0,8,f_1]\bigr \}. \end{aligned}$$

Then for \(k\in [1,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{12}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_1 \mapsto f_1\) and \(j \mapsto j+1 \pmod {11}\) on the vertices.

Example 7

Let \(V \Big (K^{(3)}_{14}\Bigr ) = \mathbb {Z}_{13} \cup \{f_1\}\) and let

$$\begin{aligned} B_1 = \bigl \{ &[12,3,0,1,2,8,9,10,f_1],\, [12,7,0,1,6,4,5,3,f_1],\, [4,2,0,1,10,3,11,8,f_1], \\ & [11,5,0,2,6,3,10,7,8],\, [12,10,0,1,7,2,8,4,f_1],\, [11,8,0,2,9,3,7,12,f_1], \\ & [10,6,0,3,8,4,9,2,f_1]\bigr \}, \\ B_2= \bigl \{&[f_1,0,1,8,11,10,12,2,3,7],\, [f_1,0,4,1,6,2,8,3,5,9],\, [0,1,2,3,12,4,5,6,8,10], \\ & [0,1,11,2,10,3,6,4,7,12],\, [0,1,6,2,8,5,11,3,4,10],\, [0,1,8,3,4,6,10,5,9,11], \\ & [0,1,3,4,8,6,9,5,7,10]\bigr \}, \\ B_3= \bigl \{&[3,7,0,4,f_1,5,10,12,1,8],\, [2,6,0,3,f_1,7,9,12,5,10],\, [4,8,0,3,9,11,12,f_1,1,7], \\ & [0,3,8,10,5,6,2,11,12,7],\, [1,2,0,3,6,7,10,11,12,5],\, [1,8,0,2,7,5,3,9,10,f_1], \\ & [2,5,0,1,6,7,9,3,4,8]\bigr \}, \\ B_4=\bigl \{&[0,1,2,3,5,6,7,10,f_1,4],\, [0,1,5,6,11,7,9,12,f_1,4],\, [0,1,7,10,2,6,9,12,f_1,3], \\ & [0,1,8,9,4,12,2,3,f_1,6],\, [0,1,11,7,9,6,10,2,f_1,4],\, [0,2,6,8,1,12,5,9,f_1,10], \\ & [0,2,7,9,3,4,6,1,5,11]\bigr \}, \\ B_5= \bigl \{&[0,1,2,4,7,11,f_1,3,5,6,9], [0,2,6,1,10,9,12,3,11,f_1,7], [3,8,2,f_1,0,1,4,9,6,5,12], \\ & [8,4,3,0,6,1,2,11,f_1,5,12], [11,4,3,0,9,5,6,8,12,2,f_1], [1,f_1,2,4,11,3,6,8,0,10,12], \\ & [5,3,1,0,9,2,8,10,4,6,12]\bigr \}. \end{aligned}$$

Then for \(k\in [1,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{14}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_1 \mapsto f_1\) and \(j \mapsto j+1 \pmod {13}\) on the vertices.

Example 8

Let \(V \Big (K^{(3)}_{16}\Bigr ) = \mathbb {Z}_{14} \cup \{f_1, f_2\}\) and let

$$\begin{aligned} B_1 = \bigl \{&[5,3,f_1,1,2,4,7,11,f_2],\, [5,3,f_2,1,2,4,7,11,f_1],\, [7,1,f_1,0,5,13,f_2,2,8], \\ & [13,5,0,1,4,7,8,9,2],\, [13,10,0,1,9,2,3,5,7],\, [12,4,0,2,5,6,8,13,1], \\ & [12,8,0,2,7,3,5,10,13],\, [7,3,0,11,8,6,9,13,5],\, [13,9,0,1,5,7,11,6,f_2] \bigr \}, \\ B'_1=\bigl \{&H[1,2,3,4,6,7,12,5,f_1],\, [2,3,4,5,7,8,13,6,f_1],\, [3,4,5,6,8,9,0,7,f_1], \\ & [4,5,6,7,9,10,1,8,f_1],\, [5,6,7,8,10,11,2,9,f_1],\, [6,7,8,9,11,12,3,10,f_1], \\ & [7,8,9,10,12,13,4,11,f_1],\, [8,9,10,11,13,0,5,12,f_2],\, [9,10,11,12,0,1,6,13,f_2], \\ & [10,11,12,13,1,2,7,0,f_2],\, [11,12,13,0,2,3,8,1,f_2],\, [12,13,0,1,3,4,9,2,f_2], \\ & [13,0,1,2,4,5,10,3,f_2],\, [0,1,2,3,5,6,11,4,f_2]\bigr \}, \\ B_2= \bigl \{& [f_1,0,1,2,4,5,8,9,10,12],\, [f_2,0,1,2,4,5,8,9,10,13],\, [f_1,4,10,5,9,6,11,13,7,8], \\ & [f_2,4,10,5,9,6,11,1,7,8],\, [0,1,9,2,8,3,7,4,10,13],\, [0,1,5,2,6,3,9,4,8,12], \\ & [0,1,7,2,9,10,12,4,8,13],\, [0,1,11,2,7,3,10,6,8,9],\, [0,1,10,2,11,4,6,7,f_1,f_2]\bigr \},\\ B'_2=\bigl \{& [0,f_1,7,1,2,3,6,8,10,13],\, [1,f_1,8,2,3,4,7,9,11,0],\, [2,f_1,9,3,4,5,8,10,12,1], \\ & [3,f_1,10,4,5,6,9,11,13,2],\, [4,f_1,11,5,6,7,10,12,0,3],\, [5,f_1,12,6,7,8,11,13,1,4], \\ & [6,f_1,13,7,8,9,12,0,2,5],\, [7,f_2,0,8,9,10,13,1,3,6],\, [8,f_2,1,9,10,11,0,2,4,7], \\ & [9,f_2,2,10,11,12,1,3,5,8],\, [10,f_2,3,11,12,13,2,4,6,9],\, [11,f_2,4,12,13,0,3,5,7,10], \\ & [12,f_2,5,13,0,1,4,6,8,11],\, [13,f_2,6,0,1,2,5,7,9,12] \bigr \},\\ B_3= \bigl \{&[3,9,0,10,1,5,8,f_2,6,12],\, [f_1,13,1,0,6,7,11,2,3,5],\, [10,13,f_1,1,f_2,4,6,3,8,12], \\ & [1,12,9,2,0,4,8,f_2,3,6],\, [13,2,1,0,8,5,9,3,f_1,7],\, [10,2,0,f_1,1,5,f_2,4,7,9], \\ & [0,5,f_1,7,1,4,8,11,12,f_2],\, [9,5,3,0,8,7,2,11,12,4],\, [2,f_2,11,6,4,0,1,5,7,13]\bigr \}, \\ B_3'= \bigl \{&[7,f_1,0,1,2,4,6,3,5,8],\, [8,f_1,1,2,3,5,7,4,6,9],\, [9,f_1,2,3,4,6,8,5,7,10], \\ & [10,f_1,3,4,5,7,9,6,8,11],\, [11,f_1,4,5,6,8,10,7,9,12],\, [12,f_1,5,6,7,9,11,8,10,13], \\ & [13,f_1,6,7,8,10,12,9,11,0],\, [0,f_2,7,8,9,11,13,10,12,1],\, [1,f_2,8,9,10,12,0,11,13,2], \\ & [2,f_2,9,10,11,13,1,12,0,3],\, [3,f_2,10,11,12,0,2,13,1,4],\, [4,f_2,11,12,13,1,3,0,2,5], \\ & [5,f_2,12,13,0,2,4,1,3,6],\, [6,f_2,13,0,1,3,5,2,4,7]\bigr \},\\ B_4=\bigl \{&[0,1,2,3,5,6,7,10,f_1,13],\, [0,1,5,6,f_2,7,8,13,2,9],\, [0,1,7,9,11,13,2,4,f_2,6], \\ & [0,1,8,f_1,12,2,4,7,f_2,13],\, [0,f_1,f_2,1,6,2,3,12,4,10],\, [0,1,11,12,f_1,2,4,8,10,3], \\ & [0,3,6,8,f_1,2,9,12,f_2,1],\, [0,1,12,f_1,3,2,4,9,f_2,13],\, [0,2,10,13,4,7,12,3,f_1,9]\bigr \},\\ B'_4= \bigl \{&[7,f_1,0,1,9,6,10,2,5,11],\, [8,f_1,1,2,10,7,11,3,6,12],\, [9,f_1,2,3,11,8,12,4,7,13], \\ & [10,f_1,3,4,12,9,13,5,8,0],\, [11,f_1,4,5,13,10,0,6,9,1],\, [12,f_1,5,6,0,11,1,7,10,2], \\ & [13,f_1,6,7,1,12,2,8,11,3],\, [0,f_2,7,8,2,13,3,9,12,4],\, [1,f_2,8,9,3,0,4,10,13,5], \\ & [2,f_2,9,10,4,1,5,11,0,6],\, [3,f_2,10,11,5,2,6,12,1,7],\, [4,f_2,11,12,6,3,7,13,2,8], \\ & [5,f_2,12,13,7,4,8,0,3,9],\, [6,f_2,13,0,8,5,9,1,4,10]\bigr \}, \\ B_{5}= \bigl \{& [1,5,13,4,9,0,f_1,f_2,2,3,6],\, [0,f_2,6,f_1,12,13,8,7,1,5,9], \\ & [0,f_2,5,f_1,10,6,8,9,1,4,11],\, [0,f_2,4,f_1,8,2,5,7,9,11,13], \\ & [0,f_2,3,f_1,6,4,17,13,8,11,12],\, [0,f_2,2,f_1,4,5,8,13,6,10,11], \\ & [0,f_2,1,f_1,2,3,6,10,5,7,12],\, [0,3,6,7,13,2,4,10,9,11,5], \\ & [5,0,1,2,9,3,4,12,6,11,13]\bigr \},\\ B'_{5}= \bigl \{& [2,1,0,f_1,7,3,4,6,8,10,13],\, [3,2,1,f_1,8,4,5,7,9,11,0], \\ & [4,3,2,f_1,9,5,6,8,10,12,1],\, [5,4,3,f_1,10,6,7,9,11,13,2], \\ & [6,5,4,f_1,11,7,8,10,12,0,3],\, [7,6,5,f_1,12,8,9,11,13,1,4], \\ & [8,7,6,f_1,13,9,10,12,0,2,5],\, [9,8,7,f_2,0,10,11,13,1,3,6], \\ & [10,9,8,f_2,1,11,12,0,2,4,7],\, [11,10,9,f_2,2,12,13,1,3,5,8], \\ & [12,11,10,f_2,3,13,0,2,4,6,9],\, [13,12,11,f_2,4,0,1,3,5,7,10], \\ & [0,13,12,f_2,5,1,2,4,6,8,11],\, [1,0,13,f_2,6,2,3,5,7,9,12] \bigr \}. \end{aligned}$$

Then for \(k\in [1,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{16}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in \{1,2\}\), and \(j \mapsto j+1 \pmod {14}\) on the vertices along with the \(H_k\)-blocks in \(B'_k\).

Example 9

Let \(V \Big (K^{(3)}_{17}\Bigr ) = \mathbb {Z}_{17}\) and let

$$\begin{aligned} B_2= \bigl \{&[0,3,13,4,12,5,11,7,8,9],\, [0,3,11,4,8,5,10,6,7,9],\, [0,1,11,2,4,3,10,5,7,15], \\ & [0,2,13,3,12,4,10,5,6,9],\, [0,1,12,2,11,3,9,5,6,10],\, [0,1,7,2,5,3,6,8,11,16], \\ & [0,1,9,2,8,4,11,5,7,14],\, [0,1,6,2,12,4,9,8,10,14],\, [0,14,16,4,5,1,10,6,9,11], \\ & [0,1,8,2,7,3,4,9,12,16]\bigr \},\\ B_3 = \bigl \{&[1,2,0,3,6,7,10,8,11,16],\, [1,8,0,2,7,5,3,4,6,14],\, [2,5,0,1,6,7,9,3,4,8], \\ & [4,8,0,9,3,5,14,1,6,12],\, [0,12,2,13,6,7,16,8,11,15],\, [0,1,15,3,4,8,13,2,5,12], \\ & [0,4,10,15,5,7,11,8,12,3],\, [0,1,14,15,4,2,11,3,6,16],\, [0,3,12,1,4,2,15,8,10,5], \\ & [0,1,13,14,7,9,15,3,4,12]\bigr \}, \\ B_4=\bigl \{&[0,1,2,3,5,6,7,10,11,15], \, [0,1,6,7,13,16,3,12,15,5],\, [0,1,8,10,5,2,4,11,15,6], \\ & [0,1,9,11,4,13,3,8,12,2],\, [0,1,10,12,14,9,2,5,8,11],\, [0,1,11,14,6,2,7,13,15,3], \\ & [0,1,12,16,5,14,3,6,8,2],\, [0,1,13,15,2,16,3,6,8,14],\, [0,1,14,2,11,8,3,5,7,15], \\ & [0,1,15,2,7,3,6,12,4,10]\bigr \},\\ B_5=\bigl \{&[1,2,0,3,16,4,6,12,7,10,13],\, [1,3,0,4,16,5,7,14,6,9,13], \\ & [1,9,0,5,16,6,8,13,7,10,15],\, [1,11,0,6,16,2,4,12,3,7,10], \\ & [1,10,0,7,16,2,4,13,5,11,14],\, [1,12,0,4,5,7,9,13,3,11,14], \\ & [1,14,0,2,4,3,8,10,5,9,13],\, [1,15,0,2,13,3,8,16,4,9,14], \\ & [2,5,0,3,10,4,8,14,1,6,12],\, [2,14,0,4,11,1,5,13,3,6,15] \bigr \}. \end{aligned}$$

Then for \(k\in [2,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{17}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(j \mapsto j+1 \pmod {17}\) on the vertices.

Example 10

Let \(V \Big (K^{(3)}_{18}\Bigr ) = \mathbb {Z}_{17} \cup \{f_1\}\) and let

$$\begin{aligned} B_5=\bigl \{&[0,1,f_1,2,4,6,8,14,7,10,13],\, [0,6,f_1,2,7,3,5,12,8,11,15], \\ & [1,9,0,5,16,6,8,13,7,10,15],\, [1,11,0,6,16,2,4,12,3,7,10], \\ & [1,10,0,7,16,2,4,13,5,11,14],\, [1,12,0,4,5,7,9,13,3,11,14], \\ & [1,14,0,2,4,3,8,10,5,9,13],\, [1,15,0,2,13,3,8,16,4,9,14], \\ & [2,5,0,3,10,4,8,14,1,6,12],\, [2,14,0,4,11,1,5,13,3,6,15], \\ & [0,3,f_1,1,5,6,7,10,12,13,14],\, [0,8,f_1,6,16,9,10,14,12,13,15] \bigr \}. \end{aligned}$$

Then an \(H_5\)-decomposition of \(K^{(3)}_{18}\) consists of the orbits of the \(H_5\)-blocks in \(B_5\) under the action of the map \(f_1 \mapsto f_1\) and \(j \mapsto j+1 \pmod {17}\) on the vertices.

Example 11

Let \(V \Big (K^{(3)}_{2,\overline{8},\overline{8}}\cup L_{\overline{8},\overline{8}}\Bigr ) = \{f_1,f_2\} \cup \mathbb {Z}_{16}\) with the vertex partition \(\bigl \{ \{f_1,f_2\},\) \(\{0,2,\ldots ,14\},\) \(\{1,3,\ldots ,15\} \bigr \}\). Now, let

$$\begin{aligned} B_3 {=} \bigl \{& [1,3,0,5,10,15,12,7,11,4],\, [1,5,0,4,11,13,14,6,12,15],\, [3,6,0,4,9,12,13,5,10,11], \\ & [2,7,6,11,3,0,12,1,8,10],\, [0,3,11,5,12,9,14,6,7,10],\, [0,3,9,1,8,10,15,4,5,12], \\ & [4,f_1,11,12,f_2,5,10,0,1,2], [14,15,f_1,10,5,f_2,12,0,1,6], [f_1,9,6,f_2,3,1,12,4,5,11]\bigr \},\\ B_5{=}\bigl \{&[1,3,0,5,10,6,9,13,2,4,7],\, [1,5,0,4,11,7,9,10,6,12,15], \\ & [3,6,0,4,9,7,10,11,8,13,14],\, [0,12,3,6,11,1,2,13,5,7,14], \\ & [0,3,11,1,2,10,13,15,4,5,8],\, [0,3,9,1,8,5,13,14,4,6,11], \\ & [4,5,f_1,10,15,0,9,14,6,7,f_2],\, [0,3,f_1,8,15,f_2,7,12,4,5,11], \\ & [4,11,f_2,10,13,0,1,2,6,7,12] \bigr \}. \end{aligned}$$

Then for \(k\in \{3,5\}\) an \(H_k\)-decomposition of \(K^{(3)}_{2,\overline{8},\overline{8}} \cup L^{(3)}_{\overline{8},\overline{8}}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in \{1,2\}\), and \(j \mapsto j+1 \pmod {16}\) on the vertices.

Example 12

Let \(V \Big (K^{(3)}_{2,8,8}\Bigr ) = \{f_1,f_2\} \cup \mathbb {Z}_{16}\) with vertex partition \(\bigl \{ \{f_1,f_2\},\) \(\{0,2,\ldots ,14\},\) \(\{1,3,\ldots ,15\} \bigr \}\). Now, let

$$\begin{aligned} B_1 = \bigl \{&[0,1,f_1,2,5,3,8,15,f_2],\, [0,1,f_2,2,5,3,8,15,f_1]\bigr \}, \\ B_2= \bigl \{& [f_1,0,1,2,5,4,9,f_2,6,13],\, [f_2,0,1,2,5,4,9,f_1,6,13]\bigr \}, \\ B_4=\bigl \{&[0,1,f_1,2,5,4,9,f_2,6,13],\, [0,1,f_2,2,5,4,9,f_1,6,13]\bigr \}. \end{aligned}$$

Then for \(k\in \{1,2,4\}\) an \(H_k\)-decomposition of \(K^{(3)}_{2,8,8}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in \{1,2\}\), and \(j \mapsto j+1 \pmod {16}\) on the vertices.

Example 13

Let \(V \Big (K^{(3)}_{4,8,8}\Bigr ) = \{f_1,f_2,f_3,f_4\} \cup \mathbb {Z}_{16}\) with the vertex partition \(\bigl \{ \{f_1,f_2,f_3,f_4\},\) \(\{0,2,\ldots ,14\},\) \(\{1,3,\ldots ,15\} \bigr \}\). Now, let

$$\begin{aligned} B_3 = \bigl \{& [0,f_1,4,f_2,1,f_3,2,3,f_4,5],\, [0,f_2,4,f_3,1,f_4,2,3,f_1,5], \\ & [0,f_3,4,f_4,1,f_1,2,3,f_2,5],\, [0,f_4,4,f_1,1,f_2,2,3,f_3,5] \bigr \}, \\ B_5=\bigl \{& [0,f_1,4,f_2,1,2,f_3,3,5,f_4,7],\, [0,f_2,4,f_3,1,2,f_4,3,5,f_1,7], \\ & [0,f_3,4,f_4,1,2,f_1,3,5,f_2,7],\, [0,f_4,4,f_1,1,2,f_2,3,5,f_3,7] \bigr \}. \end{aligned}$$

Then for \(k\in \{3,5\}\) an \(H_k\)-decomposition of \(K^{(3)}_{4,8,8}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,4]\), and \(j \mapsto j+1 \pmod {16}\) on the vertices.

Example 14

Let \(V \Big (K^{(3)}_{6,8,8}\Bigr ) = \{f_1,f_2,f_3,f_4,f_5,f_6\} \cup \mathbb {Z}_{16}\) with vertex partition \(\bigl \{ \{f_1,f_2,\ldots ,f_6\},\) \(\{0,2,\ldots ,14\},\) \(\{1,3,\ldots ,15\} \bigr \}\). Now, let

$$\begin{aligned} B_3 = \bigl \{& [0,f_1,4,f_2,1,f_3,2,3,f_4,5],\, [0,f_2,4,f_3,1,f_4,2,3,f_5,5],\, [0,f_3,4,f_4,1,f_5,2,3,f_6,5], \\ & [0,f_4,4,f_5,1,f_6,2,3,f_1,5],\, [0,f_5,4,f_6,1,f_1,2,3,f_2,5],\, [0,f_6,4,f_1,1,f_2,2,3,f_3,5] \bigr \}, \\ B_5=\bigl \{& [0,f_1,4,f_2,1,2,f_3,3,5,f_4,7],\, [0,f_2,4,f_3,1,2,f_4,3,5,f_5,7], \\ & [0,f_3,4,f_4,1,2,f_5,3,5,f_6,7],\, [0,f_4,4,f_5,1,2,f_6,3,5,f_1,7], \\ & [0,f_5,4,f_6,1,2,f_1,3,5,f_2,7],\, [0,f_6,4,f_1,1,2,f_2,3,5,f_3,7] \bigr \}. \end{aligned}$$

Then for \(k\in \{3,5\}\) an \(H_k\)-decomposition of \(K^{(3)}_{6,8,8}\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,6]\), and \(j \mapsto j+1 \pmod {16}\) on the vertices.

Example 15

Let \(V \Big (K^{(3)}_{14}\setminus K^{(3)}_6\Bigr ) = \mathbb {Z}_{8} \cup \{f_1,f_2,f_3,f_4,f_5,f_6\}\) with \( \{f_1,f_2,f_3,f_4,f_5,f_6\}\) as the set of vertices in the hole. Now, let

$$\begin{aligned} B_1 = \bigl \{& [7,5,0,1,4,2,f_1,3,6], [1,6,f_2,3,5,f_3,4,f_5,f_6], [1,4,f_3,0,2,5,f_4,f_2,6], \\ & [0,1,f_5,2,4,3,f_1,f_2,5], [0,1,f_6,2,4,3,f_1,f_3,5], [0,3,f_5,f_3,1,f_2,4,5,f_4], \\ & [0,3,f_6,f_3,1,f_2,2,f_1,f_4], [3,6,0,2,4,f_6,f_4,f_5,1]\bigr \},\\ B_{1}'= \bigl \{&[0,1,f_3,2,3,4,5,f_2,6],\, [0,1,f_2,2,3,4,5,f_1,6], \, [0,1,f_1,2,3,4,5,f_3,6],\\ & [1,2,f_3,3,4,6,7,f_2,0], \, [1,2,f_2,3,4,6,7,f_1,0],\, [1,2,f_1,3,4,6,7,f_3,0], \\ & [6,2,5,1,f_2,7,4,0,f_1],\, [7,3,6,2,f_2,0,5,1,f_1], [0,4,7,3,f_2,1,6,2,f_1],\\ & [1,5,0,4,f_2,2,7,3,f_1], \, [2,6,1,5,f_4,3,0,4,f_3],\, [3,7,2,6,f_4,4,1,5,f_3], \\ & [4,0,3,7,f_4,5,2,6,f_3],\, [5,1,4,0,f_4,6,3,7,f_3], \, [6,5,4,0,f_5,2,f_4,0,3],\\ & [7,6,5,1,f_5,3,f_4,1,4], \, [0,7,6,2,f_5,4,f_4,2,5],\, [1,0,7,3,f_5,5,f_4,3,6], \\ & [2,1,0,4,f_6,6,f_4,4,7],\, [3,2,1,5,f_6,7,f_4,5,0], \, [4,3,2,6,f_6,0,f_4,6,1],\\ & [5,4,3,7,f_6,1,f_4,7,2]\bigr \},\\ B_2 = \bigl \{& [f_5,0,1,2,4,3,6,7,f_6,f_1],\, [f_6,6,7,3,5,1,4,2,f_5,f_2], \, [f_3,2,4,3,6,1,f_4,f_2,5,0],\\ & [f_4,0,1,2,4,3,6,f_3,f_5,5], \, [f_2,0,2,4,f_4,7,f_3,3,5,6],\, [f_1,0,3,1,f_4,2,f_5,4,5,6], \\ & [2,4,6,f_6,f_2,f_1,0,7,f_5,f_4],\, [f_6,f_5,3,f_4,4,f_3,6,2,f_2,f_1]\bigr \},\\ B_{2}'= \bigl \{&[7,f_4,3,1,6,2,5,0,4,f_3],\, [0,f_4,4,2,7,3,6,1,5,f_3], \, [1,f_4,5,3,0,4,7,2,6,f_3],\\ & [2,f_4,6,4,1,5,0,3,7,f_3], \, [3,f_2,7,5,2,6,1,4,0,f_1],\, [4,f_2,0,6,3,7,2,5,1,f_1], \\ & [5,f_2,1,7,4,0,3,6,2,f_1],\, [6,f_2,2,0,5,1,4,7,3,f_1], \, [4,5,1,7,3,f_5,0,f_1,6,f_3], \\ & [5,6,2,0,4,f_5,1,f_1,7,f_3],\, [6,7,3,1,5,f_5,2,f_1,0,f_3],\, [7,0,4,2,6,f_5,3,f_1,1,f_3], \\ & [0,1,5,3,7,f_6,4,f_1,2,f_3],\, [1,2,6,4,0,f_6,5,f_1,3,f_3], \, [2,3,7,5,1,f_6,6,f_1,4,f_3], \\ & [3,4,0,6,2,f_6,7,f_1,5,f_3],\, [f_1,0,1,2,3,4,5,f_2,6,7],\, [f_2,0,1,2,3,4,5,f_3,6,7], \\ & [f_3,0,1,2,3,4,5,f_1,6,7],\, [f_1,1,2,3,4,5,6,f_2,0,7], \, [f_2,1,2,3,4,5,6,f_3,0,7], \\ & [f_3,1,2,3,4,5,6,f_1,0,7]\bigr \},\\ B_3 = \bigl \{&[0,2,f_1,1,4,7,f_2,3,5,f_3],\, [0,2,f_4,1,4,7,f_3,3,5,f_2],\, [3,5,7,f_4,f_3,2,f_2,0,1,4], \\ & [0,1,f_5,4,2,5,f_6,f_1,f_4,7],\, [0,1,f_6,4,2,5,f_5,f_2,f_1,7],\, [f_1,2,f_6,f_2,3,f_3,f_5,0,1,6], \\ & [0,f_5,f_4,5,6,f_3,f_6,2,4,7],\, [f_1,2,f_5,f_2,4,f_4,f_6,0,1,3\bigr \}, \end{aligned}$$
$$\begin{aligned} B_3' = \bigl \{& [0,1,f_1,2,3,4,f_2,5,6,f_3],\, [1,2,f_1,3,4,5,f_2,6,7,f_3],\, [0,1,f_2,2,3,4,f_3,5,6,f_1], \\ & [1,2,f_3,0,7,6,f_2,4,5,f_1],\, [2,3,f_3,1,0,7,f_1,5,6,f_2],\, [1,2,f_2,0,7,6,f_1,4,5,f_3], \\ & [2,3,7,6,5,1,f_3,0,4,f_2],\, [3,4,0,7,6,2,f_3,1,5,f_2],\, [4,5,1,0,7,3,f_3,2,6,f_2], \\ & [5,6,2,1,0,4,f_3,3,7,f_2],\, [6,7,3,2,1,5,f_4,4,0,f_1],\, [7,0,4,3,2,6,f_4,5,1,f_1], \\ & [0,1,5,4,3,7,f_4,6,2,f_1],\, [1,2,6,5,4,0,f_4,7,3,f_1],\, [0,4,f_5,f_6,5,f_1,f_3,f_2,f_4,1], \\ & [1,5,f_5,f_6,6,f_1,f_3,f_2,f_4,2],\, [2,6,f_5,f_6,7,f_1,f_3,f_2,f_4,3], \\ & [3,7,f_5,f_6,0,f_1,f_3,f_2,f_4,4],\, [0,4,f_6,f_5,1,f_1,f_3,f_2,f_4,5], \\ & [1,5,f_6,f_5,2,f_1,f_3,f_2,f_4,6],\, [2,6,f_6,f_5,3 ,f_1,f_3,f_2,f_4,7], \\ & [3,7,f_6,f_5,4,f_1,f_3,f_2,f_4,0]\bigr \},\\ B_4= \bigl \{&[0,f_3,f_1,1,3,7,f_2,f_4,4,6], \, [3,0,1,2,5,6,f_3,4,7,f_2],\, [0,1,f_5,2,4,6,7,f_6,3,5], \\ & [f_1,0,f_5,f_4,1,f_6,f_3,2,4,f_2],\, [f_1,0,f_6,f_4,2,f_2,f_5,1,3,4], \\ & [0,f_4,f_3,f_2,1,2,f_5,f_6,3,6],\, [0,f_4,3,f_2,f_6,7,f_3,f_5,1,4]\bigr \},\\ B'_4=\bigl \{&[0,1,f_1,2,3,4,5,f_2,6,7],\, [0,1,f_2,2,3,4,5,f_3,6,7],\, [0,1,f_3,2,3,4,5,f_1,6,7], \\ & [1,2,f_1,3,4,5,6,f_2,0,7],\, [1,2,f_2,3,4,5,6,f_3,0,7],\, [1,2,f_3,3,4,5,6,f_1,0,7], \\ & [f_1,0,4,5,6,f_3,7,2,f_4,1],\, [f_1,1,5,6,7,f_3,0,3,f_4,2],\, [f_1,2,6,7,0,f_3,1,4,f_4,3], \\ & [f_1,3,7,0,1,f_3,2,5,f_4,4],\, [f_2,4,0,1,2,f_3,3,6,f_4,5],\, [f_2,5,1,2,3,f_3,4,7,f_4,6], \\ & [f_2,6,2,3,4,f_3,5,0,f_4,7],\, [f_2,7,3,4,5,f_3,6,1,f_4,0],\, [f_3,0,4,7,2,f_1,f_2,1,3,5], \\ & [f_3,1,5,0,3,f_1,f_2,2,4,6],\, [f_3,2,6,1,4,f_1,f_2,3,5,7],\, [f_3,3,7,2,5,f_1,f_2,4,6,0], \\ & [f_4,4,0,3,6,f_1,f_2,5,7,1],\, [f_4,5,1,4,7,f_1,f_2,6,0,2],\, [f_4,6,2,5,0,f_1,f_2,7,1,3], \\ & [f_4,7,3,6,1,f_1,f_2,0,2,4],\, [f_5,4,0,1,5,7,f_4,f_1,3,6],\, [f_5,5,1,2,6,0,f_4,f_1,4,7], \\ & [f_5,6,2,3,7,1,f_4,f_1,5,0],\, [f_5,7,3,4,0,2,f_4,f_1,6,1],\, [f_6,0,4,5,1,3,f_4,f_1,7,2], \\ & [f_6,1,5,6,2,4,f_4,f_1,0,3],\, [f_6,2,6,7,3,5,f_4,f_1,1,4],\, [f_6,3,7,0,4,6,f_4,f_1,2,5]\bigr \},\\ B_{5} = \bigl \{&[6,f_4,f_6,f_5,7,1,3,f_3,0,2,5],\, [5,f_3,f_4,f_5,7,f_2,3,6,0,2,4], \\ & [f_5,1,f_3,5,f_6,3,4,6,f_1,0,2],\, [f_5,2,f_2,f_6,5,3,6,f_4,0,1,4], \\ & [f_3,1,f_2,f_4,4,2,5,6,f_1,0,3],\, [f_5,6,f_1,f_6,7,f_4,4,5,f_3,0,3], \\ & [3,f_2,f_1,f_4,5,2,4,f_6,f_5,0,1],\, [0,2,f_5,3,6,f_4,5,7,f_6,1,4]\bigr \},\\ B'_{5} = \bigl \{&[0,1,f_1,2,3,f_2,4,5,f_3,6,7],\, [0,1,f_2,2,3,f_3,4,5,f_1,6,7],\, [0,1,f_3,2,3,f_1,4,5,f_2,6,7], \\ & [1,2,f_1,3,4,f_2,5,6,f_3,7,0],\, [1,2,f_2,3,4,f_3,5,6,f_1,7,0],\, [1,2,f_3,3,4,f_1,5,6,f_2,7,0], \\ & [1,2,0,f_1,4,f_2,3,7,f_6,5,6],\, [2,3,1,f_1,5,f_2,4,0,f_6,6,7],\, [3,4,2,f_1,6,f_2,5,1,f_6,7,0], \\ & [4,5,3,f_1,7,f_2,6,2,f_6,0,1],\, [5,6,4,f_3,0,f_4,7,3,f_6,1,2],\, [6,7,5,f_3,1,f_4,0,4,f_6,2,3], \\ & [7,0,6,f_3,2,f_4,1,5,f_6,3,4],\, [0,1,7,f_3,3,f_4,2,6,f_6,4,5], \\ & [1,3,4,0,f_5,f_2,5,7,f_1,f_3,2],\, [2,4,5,1,f_5,f_2,6,0,f_1,f_3,3], \\ & [3,5,6,2,f_5,f_2,7,1,f_1,f_3,4],\, [4,6,7,3,f_5,f_2,0,2,f_1,f_3,5], \\ & [5,7,0,4,f_6,f_2,1,3,f_1,f_3,6],\, [6,0,1,5,f_6,f_2,2,4,f_1,f_3,7], \\ & [7,1,2,6,f_6,f_2,3,5,f_1,f_3,0],\, [0,2,3,7,f_6,f_2,4,6,f_1,f_3,1]\bigr \}. \end{aligned}$$

Then for \(k\in [1,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{14}\setminus K^{(3)}_6\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,6]\), and \(j \mapsto j+1 \pmod {8}\) on the vertices along with the \(H_k\)-blocks in \(B'_k\).

Example 16

Let \(V \Big (K^{(3)}_{16}\setminus K^{(3)}_8\Bigr ) = \mathbb {Z}_{8} \cup \{f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8\}\) with the set \(\{f_1, \dots , f_8\}\) as the set of vertices in the hole. Now, let

$$\begin{aligned} B_1 = \bigl \{& [7,5,0,1,4,2,f_1,3,6],\, [0,6,f_2,2,5,f_3,4,f_5,f_6],\, [1,4,f_3,0,2,5,f_4,f_2,6], \\ & [0,1,f_5,2,4,3,f_1,f_2,5],\, [0,1,f_6,2,4,3,f_1,f_3,5],\, [0,3,f_5,f_3,1,f_2,4,5,f_4], \\ & [5,6,f_7,1,3,f_1,0,2,4],\, [0,1,f_8,2,4,3,f_1,f_4,5],\, [f_3,0,f_8,f_2,1,2,f_4,f_6,3], \\ & [0,3,f_7,f_3,1,2,f_2,f_6,4],\, [0,3,f_8,f_5,1,2,f_6,4,7], \, [f_5,0,f_7,f_4,1,2,f_6,f_3,3]\bigr \},\\ B_1'= \bigl \{&[0,1,f_3,2,3,4,5,f_2,6],\, [0,1,f_2,2,3,4,5,f_1,6],\, [0,1,f_1,2,3,4,5,f_3,6], \\ & [1,2,f_3,3,4,6,7,f_2,0],\, [1,2,f_2,3,4,6,7,f_1,0],\, [1,2,f_1,3,4,6,7,f_3,0], \\ & [6,2,5,1,f_2,7,4,0,f_1],\, [7,3,6,2,f_2,0,5,1,f_1],\, [0,4,7,3,f_2,1,6,2,f_1], \\ & [1,5,0,4,f_2,2,7,3,f_1],\, [2,6,1,5,f_4,3,0,4,f_3],\, [3,7,2,6,f_4,4,1,5,f_3], \\ & [4,0,3,7,f_4,5,2,6,f_3],\, [5,1,4,0,f_4,6,3,7,f_3],\, [6,5,4,0,f_5,2,f_4,0,3], \\ & [7,6,5,1,f_5,3,f_4,1,4],\, [0,7,6,2,f_5,4,f_4,2,5],\, [1,0,7,3,f_5,5,f_4,3,6], \\ & [2,1,0,4,f_6,6,f_4,4,7],\, [3,2,1,5,f_6,7,f_4,5,0],\, [4,3,2,6,f_6,0,f_4,6,1], \\ & [5,4,3,7,f_6,1,f_4,7,2],\, [1,7,4,f_5,f_4,0,f_7,f_8,2],\, [2,0,5,f_5,f_4,1,f_7,f_8,3], \\ & [3,1,6,f_5,f_4,2,f_7,f_8,4],\, [4,2,7,f_5,f_4,3,f_7,f_8,5],\, [5,3,0,f_5,f_4,4,f_8,f_7,6], \\ & [6,4,1,f_5,f_4,5,f_8,f_7,7],\, [7,5,2,f_5,f_4,6,f_8,f_7,0],\, [0,6,3,f_5,f_4,7,f_8,f_7,1]\bigr \},\\ B_2 = \bigl \{& [f_8,f_3,2,f_2,1,f_1,0,3,5,f_5],\, [f_6,0,1,2,4,3,6,5,f_5,f_1],\, [f_7,f_4,2,f_3,1,f_2,0,3,4,f_5], \\ & [f_7,0,1,2,4,3,6,5,f_6,f_1],\, [f_8,f_7,0,f_6,1,f_4,2,3,f_5,f_3],\, [f_6,f_5,0,f_4,1,f_3,2,f_1,7,f_7], \\ & [f_5,f_4,2,0,3,4,f_8,5,f_7,f_6],\, [f_8,0,1,2,4,3,6,5,f_2,f_6],\, [f_1,0,f_4,1,4,3,5,6,f_5,f_2], \\ & [0,1,2,4,6,f_7,f_5,5,f_3,f_1],\, [f_2,0,2,4,f_4,7,f_3,3,5,6],\, [f_4,0,1,2,4,3,6,f_1,f_2,5], \\ & [f_3,2,4,3,6,1,f_4,f_2,5,0]\bigr \}, \\ B_2' = \bigl \{ & [7,f_4,3,1,6,2,5,0,4,f_3],\, [0,f_4,4,2,7,3,6,1,5,f_3],\, [1,f_4,5,3,0,4,7,2,6,f_3], \\ & [2,f_4,6,4,1,5,0,3,7,f_3],\, [3,f_2,7,5,2,6,1,4,0,f_1],\, [4,f_2,0,6,3,7,2,5,1,f_1], \\ & [5,f_2,1,7,4,0,3,6,2,f_1],\, [6,f_2,2,0,5,1,4,7,3,f_1],\, [0,1,5,4,f_5,3,7,2,6,f_6], \\ & [1,2,6,5,f_5,4,0,3,7,f_6],\, [2,3,7,6,f_5,5,1,4,0,f_6],\, [3,4,0,7,f_5,6,2,5,1,f_6], \\ & [4,5,1,0,f_7,7,3,6,2,f_8],\, [5,6,2,1,f_7,0,4,7,3,f_8],\, [6,7,3,2,f_7,1,5,0,4,f_8], \\ & [7,0,4,3,f_7,2,6,1,5,f_8],\, [f_1,0,1,2,3,4,5,f_2,6,7],\, [f_2,0,1,2,3,4,5,f_3,6,7], \\ & [f_3,0,1,2,3,4,5,f_1,6,7],\, [f_1,1,2,3,4,5,6,f_2,0,7],\, [f_2,1,2,3,4,5,6,f_3,0,7], \\ & [f_3,1,2,3,4,5,6,f_1,0,7]\bigr \}, \\ B_3 = \bigl \{&[0,2,f_1,1,4,7,f_2,3,5,f_3],\, [0,2,f_4,1,4,7,f_3,3,5,f_2],\, [3,5,7,f_4,f_3,2,f_2,0,1,4], \\ & [0,1,f_5,4,2,5,f_6,f_1,f_4,7],\, [0,1,f_6,4,2,5,f_5,f_2,f_1,7],\, [0,f_5,f_4,5,6,f_3,f_6,2,4,7], \\ & [f_8,2,f_1,4,f_7,5,6,0,1,3],\, [f_7,0,f_2,1,f_8,2,3,f_4,f_6,4],\, [f_8,0,f_3,1,f_7,2,4,f_2,f_5,3], \\ & [f_7,1,f_4,3,f_8,2,4,0,5,7],\, [f_8,0,f_5,1,f_7,2,5,f_1,f_6,3],\, [f_7,0,f_6,1,f_8,2,5,f_3,f_5,3]\bigr \},\\ \end{aligned}$$
$$\begin{aligned} B_3' = \bigl \{&[0,1,f_1,2,3,4,f_2,5,6,f_3],\, [1,2,f_1,3,4,5,f_2,6,7,f_3],\, [0,1,f_2,2,3,4,f_3,5,6,f_1], \\ & [1,2,f_3,0,7,6,f_2,4,5,f_1],\, [2,3,f_3,1,0,7,f_1,5,6,f_2],\, [1,2,f_2,0,7,6,f_1,4,5,f_3], \\ & [2,3,7,6,5,1,f_3,0,4,f_2],\, [3,4,0,7,6,2,f_3,1,5,f_2],\, [4,5,1,0,7,3,f_3,2,6,f_2], \\ & [5,6,2,1,0,4,f_3,3,7,f_2],\, [6,7,3,2,1,5,f_4,4,0,f_1],\, [7,0,4,3,2,6,f_4,5,1,f_1], \\ & [0,1,5,4,3,7,f_4,6,2,f_1],\, [1,2,6,5,4,0,f_4,7,3,f_1],\, [0,4,f_5,f_6,5,f_1,f_3,f_2,f_4,1], \\ & [1,5,f_5,f_6,6,f_1,f_3,f_2,f_4,2],\, [2,6,f_5,f_6,7,f_1,f_3,f_2,f_4,3], \\ & [3,7,f_5,f_6,0,f_1,f_3,f_2,f_4,4],\, [0,4,f_6,f_5,1,f_1,f_3,f_2,f_4,5], \\ & [1,5,f_6,f_5,2,f_1,f_3,f_2,f_4,6],\, [2,6,f_6,f_5,3,f_1,f_3,f_2,f_4,7], \\ & [3,7,f_6,f_5,4,f_1,f_3,f_2,f_4,0],\, [0,4,f_7,f_8,1,f_1,f_5,f_2,f_6,2], \\ & [1,5,f_7,f_8,2,f_1,f_5,f_2,f_6,3],\, [2,6,f_7,f_8,3,f_1,f_5,f_2,f_6,4], \\ & [3,7,f_7,f_8,4,f_1,f_5,f_2,f_6,5],\, [0,4,f_8,f_7,5,f_1,f_5,f_2,f_6,6], \\ & [1,5,f_8,f_7,6,f_1,f_5,f_2,f_6,7],\, [2,6,f_8,f_7,7,f_1,f_5,f_2,f_6,0], \\ & [3,7,f_8,f_7,0,f_1,f_5,f_2,f_6,1]\bigr \},\\ B_4 = \bigl \{&[0,f_3,f_1,1,3,7,f_2,f_4,4,6],\, [f_1,0,f_6,f_4,2,f_2,f_5,1,3,4],\, [0,f_4,f_7,1,f_1,2,f_2,f_8,3,6], \\ & [0,f_3,f_8,1,f_5,2,f_2,f_7,3,f_6],\, [0,f_6,f_8,1,f_1,2,f_3,f_7,3,f_5],\, [0,1,f_7,2,4,6,7,f_8,3,5], \\ & [0,f_7,f_8,1,f_4,2,f_2,f_3,3,f_6],\, [0,3,f_5,1,2,4,7,f_6,5,6],\, [0,2,f_2,4,7,3,f_3,f_5,1,f_1], \\ & [0,f_4,f_5,1,f_6,2,f_7,5,7,4],\, [4,0,1,3,f_3,f_4,2,5,f_2,f_6]\bigr \},\\ B'_4=\bigl \{&[0,1,f_1,2,3,4,5,f_2,6,7],\, [0,1,f_2,2,3,4,5,f_3,6,7],\, [0,1,f_3,2,3,4,5,f_1,6,7], \\ & [1,2,f_1,3,4,5,6,f_2,0,7],\, [1,2,f_2,3,4,5,6,f_3,0,7],\, [1,2,f_3,3,4,5,6,f_1,0,7], \\ & [f_1,0,4,5,6,f_3,7,2,f_4,1],\, [f_1,1,5,6,7,f_3,0,3,f_4,2],\, [f_1,2,6,7,0,f_3,1,4,f_4,3], \\ & [f_1,3,7,0,1,f_3,2,5,f_4,4],\, [f_2,4,0,1,2,f_3,3,6,f_4,5],\, [f_2,5,1,2,3,f_3,4,7,f_4,6], \\ & [f_2,6,2,3,4,f_3,5,0,f_4,7],\, [f_2,7,3,4,5,f_3,6,1,f_4,0],\, [f_3,0,4,7,2,f_1,f_2,1,3,5], \\ & [f_3,1,5,0,3,f_1,f_2,2,4,6],\, [f_3,2,6,1,4,f_1,f_2,3,5,7],\, [f_3,3,7,2,5,f_1,f_2,4,6,0], \\ & [f_4,4,0,3,6,f_1,f_2,5,7,1],\, [f_4,5,1,4,7,f_1,f_2,6,0,2],\, [f_4,6,2,5,0,f_1,f_2,7,1,3], \\ & [f_4,7,3,6,1,f_1,f_2,0,2,4],\, [f_5,4,0,1,5,7,f_4,f_1,3,6],\, [f_5,5,1,2,6,0,f_4,f_1,4,7], \\ & [f_5,6,2,3,7,1,f_4,f_1,5,0],\, [f_5,7,3,4,0,2,f_4,f_1,6,1],\, [f_6,0,4,5,1,3,f_4,f_1,7,2], \\ & [f_6,1,5,6,2,4,f_4,f_1,0,3],\, [f_6,2,6,7,3,5,f_4,f_1,1,4],\, [f_6,3,7,0,4,6,f_4,f_1,2,5],\\ & [f_7,4,0,2,f_6,f_3,f_4,5,7,f_5],\, [f_7,5,1,3,f_6,f_3,f_4,6,0,f_5],\, [f_7,6,2,4,f_6,f_3,f_4,7,1,f_5], \\ & [f_7,7,3,5,f_6,f_3,f_4,0,2,f_5],\, [f_8,0,4,6,f_6,f_3,f_4,1,3,f_5],\, [f_8,1,5,7,f_6,f_3,f_4,2,4,f_5],\\ & [f_8,2,6,0,f_6,f_3,f_4,3,5,f_5],\, [f_8,3,7,1,f_6,f_3,f_4,4,6,f_5]\bigr \}, \\ B_{5} = \bigl \{&[0,4,1,2,6,f_2,f_3,3,5,7,f_4],\, [6,f_6,f_8,7,f_7,3,f_3,f_4,0,f_1,f_2], \\ & [7,f_6,f_7,3,6,2,5,f_2,4,f_5,f_8],\, [4,f_7,f_5,3,f_6,0,2,f_2,1,6,f_3], \\ & [1,f_7,f_4,2,f_8,0,3,5,4,6,7],\, [0,f_5,f_4,1,f_6,2,4,6,3,5,f_8], \\ & [7,f_8,f_3,6,f_7,2,f_1,f_5,0,3,f_6],\, [4,f_5,f_3,6,f_6,0,3,f_1,1,2,f_4], \\ & [5,f_8,f_2,3,f_7,2,4,f_6,0,1,f_5],\, [0,f_5,f_2,f_6,4,2,5,f_4,1,3,f_3], \\ & [0,f_3,f_1,f_4,4,6,7,f_7,1,3,f_5],\, [f_1,f_6,1,f_2,f_4,4,7,f_8,3,5,f_7], \\ & [0,f_8,f_1,1,f_7,6,7,f_6,2,5,f_5]\bigr \},\\ B'_{5}= \bigl \{& [0,1,f_1,2,3,f_2,4,5,f_3,6,7],\, [0,1,f_2,2,3,f_3,4,5,f_1,6,7],\, [0,1,f_3,2,3,f_1,4,5,f_2,6,7], \\ & [1,2,f_1,3,4,f_2,5,6,f_3,7,0],\, [1,2,f_2,3,4,f_3,5,6,f_1,7,0],\, [1,2,f_3,3,4,f_1,5,6,f_2,7,0], \\ & [5,6,7,3,f_2,1,2,f_8,0,4,f_1],\, [6,7,0,4,f_2,2,3,f_8,1,5,f_1],\, [7,0,1,5,f_2,3,4,f_8,2,6,f_1], \\ & [0,1,2,6,f_2,4,5,f_8,3,7,f_1],\, [1,2,3,7,f_3,5,6,f_8,0,4,f_4],\, [2,3,4,0,f_3,6,7,f_8,1,5,f_4], \\ & [3,4,5,1,f_3,7,0,f_8,2,6,f_4],\, [4,5,6,2,f_3,0,1,f_8,3,7,f_4],\, [1,3,0,4,f_5,5,7,f_1,2,6,f_6], \\ & [2,4,1,5,f_5,6,0,f_1,3,7,f_6],\, [3,5,2,6,f_5,7,1,f_1,4,0,f_6],\, [4,6,3,7,f_5,0,2,f_1,5,1,f_6], \\ & [5,7,4,0,f_7,1,3,f_1,6,2,f_8],\, [6,0,5,1,f_7,2,4,f_1,7,3,f_8],\, [7,1,6,2,f_7,3,5,f_1,0,4,f_8], \\ & [0,2,7,3,f_7,4,6,f_1,1,5,f_8]\bigr \} . \end{aligned}$$

Then for \(k\in [1,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{16}\setminus K^{(3)}_8\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,8]\), and \(j \mapsto j+1 \pmod {8}\) on the vertices along with the \(H_k\)-blocks in \(B'_k\).

Example 17

Let \(V \Big (K^{(3)}_{17}\setminus K^{(3)}_9\Bigr ) = \mathbb {Z}_{8} \cup \{f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8,f_9\}\) with \(\{f_1, \dots , f_9\}\) as the set of vertices in the hole. Now, let

$$\begin{aligned} B_2=\bigl \{&[f_8,f_3,2,f_2,1,f_1,0,3,5,f_5],\, [f_6,0,1,2,4,3,6,5,f_5,f_1],\, [f_7,f_4,2,f_3,1,f_2,0,3,4,f_5], \\ & [f_7,0,1,2,4,3,6,5,f_6,f_1],\, [f_8,f_7,0,f_6,1,f_4,2,3,f_5,f_3],\, [f_6,f_5,0,f_4,1,f_3,2,f_1,7,f_7], \\ & [f_8,0,1,2,4,3,6,5,f_2,f_6],\, [f_5,f_4,2,0,3,4,f_8,5,f_7,f_6],\, [0,1,2,4,6,f_7,f_5,5,f_3,f_1], \\ & [f_9,f_4,0,f_3,1,f_1,2,3,6,f_2],\, [f_2,0,f_5,1,f_4,2,f_9,3,5,6],\, [f_4,0,1,2,4,3,6,7,f_9,f_5], \\ & [f_1,0,3,1,f_4,2,f_2,4,7,f_3],\, [f_9,0,1,2,4,3,6,f_2,f_3,5],\, [f_9,f_8,7,f_7,6,f_6,5,1,f_4,f_3]\bigr \},\\ B'_2 = \bigl \{& [f_1,0,1,2,3,4,5,f_2,6,7],\, [f_2,0,1,2,3,4,5,f_3,6,7],\, [f_3,0,1,2,3,4,5,f_1,6,7], \\ & [f_1,1,2,3,4,5,6,f_2,0,7],\, [f_2,1,2,3,4,5,6,f_3,0,7],\, [f_3,1,2,3,4,5,6,f_1,0,7], \\ & [f_1,0,4,1,3,2,6,5,7,f_2],\, [f_1,0,6,2,4,3,5,1,7,f_2],\, [f_1,0,2,1,5,3,7,4,6,f_3], \\ & [f_2,0,2,1,3,4,6,5,7,f_1],\, [f_3,0,2,1,3,5,7,f_1,4,6],\, [f_2,0,6,2,4,3,5,1,7,f_3], \\ & [f_3,0,6,2,4,3,5,1,7,f_1],\, [7,f_2,3,1,6,2,5,0,4,f_3],\, [0,f_2,4,2,7,3,6,1,5,f_3], \\ & [1,f_2,5,3,0,4,7,2,6,f_3],\, [2,f_2,6,4,1,5,0,3,7,f_3],\, [3,f_4,7,5,2,6,1,4,0,f_5], \\ & [4,f_4,0,6,3,7,2,5,1,f_5],\, [5,f_4,1,7,4,0,3,6,2,f_5],\, [6,f_4,2,0,5,1,4,7,3,f_5], \\ & [0,1,5,4,f_6,3,7,2,6,f_7],\, [1,2,6,5,f_6,4,0,3,7,f_7],\, [2,3,7,6,f_6,5,1,4,0,f_7], \\ & [3,4,0,7,f_6,6,2,5,1,f_7],\, [4,5,1,0,f_8,7,3,6,2,f_9],\, [5,6,2,1,f_8,0,4,7,3,f_9], \\ & [6,7,3,2,f_8,1,5,0,4,f_9],\, [7,0,4,3,f_8,2,6,1,5,f_9]\bigr \}, \\ B_3 = \bigl \{&[0,2,f_1,1,4,7,f_2,3,5,f_3],\, [0,2,f_4,1,4,7,f_3,3,5,f_2],\, [3,5,7,f_4,f_3,2,f_2,0,1,4], \\ & [0,1,f_5,4,2,5,f_6,f_1,f_4,7],\, [0,1,f_6,4,2,5,f_5,f_2,f_1,7],\, [f_1,2,f_5,f_2,4,f_4,f_6,0,1,3], \\ & [f_1,2,f_6,f_2,3,f_3,f_5,0,1,6],\, [0,f_5,f_4,5,6,f_3,f_6,2,4,7],\, [0,1,f_7,2,4,7,f_8,f_1,f_9,6], \\ & [0,1,f_8,2,4,7,f_9,f_1,f_7,6],\, [0,1,f_9,2,4,7,f_7,f_1,f_8,6],\, [0,f_7,f_2,1,f_8,f_3,2,f_4,f_9,3], \\ & [f_2,0,f_9,1,f_3,f_7,2,f_4,f_8,3],\, [f_9,0,f_5,1,f_8,f_6,2,f_4,f_7,3], \\ & [0,f_8,f_7,1,f_9,f_6,2,f_3,3,4],\, [f_6,0,f_7,f_5,1,f_8,f_9,f_2,2,3]\bigr \},\\ B_3' = \bigl \{& [f_8,1,5,6,f_1,2,3,f_9,0,4],\, [f_7,3,7,0,f_1,4,5,f_9,2,6],\, [f_7,5,1,0,f_1,3,4,f_8,2,6], \\ & [f_1,6,7,3,f_9,1,5,f_8,0,4],\, [f_1,1,2,6,f_7,0,4,f_8,3,7],\, [2,3,7,6,5,1,f_3,0,4,f_2], \\ & [3,4,0,7,6,2,f_3,1,5,f_2],\, [4,5,1,0,7,3,f_3,2,6,f_2],\, [5,6,2,1,0,4,f_3,3,7,f_2], \\ & [6,7,3,2,1,5,f_4,4,0,f_1],\, [7,0,4,3,2,6,f_4,5,1,f_1],\, [0,1,5,4,3,7,f_4,6,2,f_1], \\ & [1,2,6,5,4,0,f_4,7,3,f_1],\, [0,4,f_5,f_6,5,f_1,f_3,f_2,f_4,1],\, [1,5,f_5,f_6,6,f_1,f_3,f_2,f_4,2], \\ & [2,6,f_5,f_6,7,f_1,f_3,f_2,f_4,3],\, [3,7,f_5,f_6,0,f_1,f_3,f_2,f_4,4], \\ & [0,4,f_6,f_5,1,f_1,f_3,f_2,f_4,5],\, [1,5,f_6,f_5,2,f_1,f_3,f_2,f_4,6], \\ & [2,6,f_6,f_5,3,f_1,f_3,f_2,f_4,7],\, [3,7,f_6,f_5,4,f_1,f_3,f_2,f_4,0]\bigr \},\\ \end{aligned}$$
$$\begin{aligned} B_4 = \bigl \{&[0,f_3,f_1,1,3,7,f_2,f_4,4,6],\, [f_1,0,f_6,f_4,2,f_2,f_5,1,3,4],\, [0,f_3,f_8,1,f_5,2,f_2,f_7,3,f_6], \\ & [0,f_6,f_8,1,f_1,2,f_3,f_7,3,f_5],\, [0,3,f_5,1,2,4,7,f_6,5,6],\, [0,1,f_7,2,4,6,7,f_8,3,5], \\ & [0,f_4,f_7,1,f_1,2,f_2,f_8,3,6]{,}\; [0,f_8,f_9,1,f_7,2,f_4,f_5,3,f_3]{,}\; [0,f_6,f_9,1,f_5,2,f_2,f_3,3,4], \\ & [0,f_4,f_9,1,3,5,f_8,f_7,4,7],\, [0,f_1,f_9,1,f_3,2,5,f_2,3,f_6],\, [0,f_5,f_6,1,f_3,5,f_9,2,4,f_2], \\ & [0,f_2,f_9,1,2,3,4,6,f_1,f_5],\, [4,0,1,3,f_3,2,5,f_4,6,f_8]\bigr \},\\ B'_4=\bigl \{&[0,7,f_1,3,4,f_2,1,2,6,f_9],\, [f_1,2,3,7,f_9,0,1,f_2,5,6],\, [1,2,f_1,6,7,f_9,0,4,5,f_2], \\ & [6,f_1,5,1,f_9,3,4,f_2,0,7],\, [0,1,f_1,4,5,2,3,f_2,6,7],\, [f_1,0,4,5,6,f_3,7,2,f_4,1], \\ & [f_1,1,5,6,7,f_3,0,3,f_4,2],\, [f_1,2,6,7,0,f_3,1,4,f_4,3],\, [f_1,3,7,0,1,f_3,2,5,f_4,4], \\ & [f_2,4,0,1,2,f_3,3,6,f_4,5],\, [f_2,5,1,2,3,f_3,4,7,f_4,6],\, [f_2,6,2,3,4,f_3,5,0,f_4,7], \\ & [f_2,7,3,4,5,f_3,6,1,f_4,0],\, [f_3,0,4,7,2,f_1,f_2,1,3,5],\, [f_3,1,5,0,3,f_1,f_2,2,4,6], \\ & [f_3,2,6,1,4,f_1,f_2,3,5,7],\, [f_3,3,7,2,5,f_1,f_2,4,6,0],\, [f_4,4,0,3,6,f_1,f_2,5,7,1], \\ & [f_4,5,1,4,7,f_1,f_2,6,0,2],\, [f_4,6,2,5,0,f_1,f_2,7,1,3],\, [f_4,7,3,6,1,f_1,f_2,0,2,4], \\ & [f_5,4,0,1,5,7,f_4,f_1,3,6],\, [f_5,5,1,2,6,0,f_4,f_1,4,7],\, [f_5,6,2,3,7,1,f_4,f_1,5,0], \\ & [f_5,7,3,4,0,2,f_4,f_1,6,1],\, [f_6,0,4,5,1,3,f_4,f_1,7,2],\, [f_6,1,5,6,2,4,f_4,f_1,0,3], \\ & [f_6,2,6,7,3,5,f_4,f_1,1,4],\, [f_6,3,7,0,4,6,f_4,f_1,2,5],\, [f_7,4,0,2,f_6,f_3,f_4,5,7,f_5], \\ & [f_7,5,1,3,f_6,f_3,f_4,6,0,f_5],\, [f_7,6,2,4,f_6,f_3,f_4,7,1,f_5],\, [f_7,7,3,5,f_6,f_3,f_4,0,2,f_5], \\ & [f_8,0,4,6,f_6,f_3,f_4,1,3,f_5],\, [f_8,1,5,7,f_6,f_3,f_4,2,4,f_5],\, [f_8,2,6,0,f_6,f_3,f_4,3,5,f_5], \\ & [f_8,3,7,1,f_6,f_3,f_4,4,6,f_5]\bigr \},\\ B_{5} = \bigl \{&[0,1,f_7,2,4,f_8,3,6,f_1,f_9,7],\, [0,1,f_8,2,4,f_9,3,6,f_1,f_7,7], \\ & [0,1,f_9,2,4,f_7,3,6,f_1,f_8,7],\, [0,f_7,f_9,f_8,1,f_3,2,3,f_2,4,5], \\ & [f_6,0,f_8,f_7,1,f_2,f_9,2,f_4,3,4],\, [f_2,0,f_7,f_3,1,f_5,f_8,2,f_6,f_9,3], \\ & [f_2,0,f_8,f_3,1,f_4,f_7,2,f_5,f_9,3],\, [f_5,0,f_7,f_6,1,f_4,f_8,2,f_3,f_9,3], \\ & [6,f_4,f_6,f_5,7,1,3,f_3,0,2,5],\, [5,f_3,f_4,f_5,7,f_2,3,6,0,2,4], \\ & [f_5,1,f_3,5,f_6,3,4,6,f_1,0,2],\, [f_5,2,f_2,f_6,5,3,6,f_4,0,1,4], \\ & [f_3,1,f_2,f_4,4,2,5,6,f_1,0,3],\, [f_5,6,f_1,f_6,7,f_4,f_9,5,f_3,0,3], \\ & [3,f_3,f_1,f_4,5,2,4,f_6,f_5,0,1],\, [0,2,f_5,3,6,f_4,5,7,f_6,1,4]\bigr \},\\ B'_{5}= \bigl \{& [1,2,f_1,5,6,f_8,0,4,f_9,3,7],\, [3,4,f_1,7,0,f_8,1,5,f_9,2,6],\, [2,3,f_1,6,7,f_7,1,5,f_9,0,4], \\ & [0,1,f_1,4,5,f_7,3,7,f_8,2,6],\, [0,4,f_7,2,6,f_9,1,5,f_8,3,7],\, [1,2,0,f_1,4,f_2,3,7,f_6,5,6], \\ & [2,3,1,f_1,5,f_2,4,0,f_6,6,7],\, [3,4,2,f_1,6,f_2,5,1,f_6,7,0],\, [4,5,3,f_1,7,f_2,6,2,f_6,0,1], \\ & [5,6,4,f_3,0,f_4,7,3,f_6,1,2],\, [6,7,5,f_3,1,f_4,0,4,f_6,2,3],\, [7,0,6,f_3,2,f_4,1,5,f_6,3,4], \\ & [0,1,7,f_3,3,f_4,2,6,f_6,4,5],\, [2,f_1,f_2,1,7,3,5,6,f_5,0,4],\, [3,f_1,f_2,2,0,4,6,7,f_5,1,5], \\ & [4,f_1,f_2,3,1,5,7,0,f_5,2,6],\, [5,f_1,f_2,4,2,6,0,1,f_5,3,7],\, [6,f_1,f_2,5,3,7,1,2,f_6,0,4], \\ & [7,f_1,f_2,6,4,0,2,3,f_6,1,5]{,}\, [0,f_1,f_2,7,5,1,3,4,f_6,2,6]{,}\, [1,f_1,f_2,0,6,2,4,5,f_6,3,7] \bigr \}. \end{aligned}$$

Then for \(k\in [2,5]\) an \(H_k\)-decomposition of \(K^{(3)}_{17}\setminus K^{(3)}_9\) consists of the orbits of the \(H_k\)-blocks in \(B_k\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,9]\), and \(j \mapsto j+1 \pmod {8}\) on the vertices along with the \(H_k\)-blocks in \(B'_k\).

Example 18

Let \(V \Big (K^{(3)}_{18}\setminus K^{(3)}_{10}\Bigr ) = \mathbb {Z}_{8} \cup \{f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8,f_9,f_{10}\}\) with \(\{f_1, \dots , f_{10}\}\) as the set of vertices in the hole. Now, let

$$\begin{aligned} B_{5} = \bigl \{&[6,f_9,f_8,7,f_{10},0,f_1,f_3,1,2,4],\, [4,f_9,f_7,5,f_{10},0,f_1,f_4,1,3,6], \\ & [2,f_6,f_7,f_8,3,4,6,f_1,0,1,5],\, [4,f_9,f_6,6,f_{10},0,2,f_3,3,5,f_2], \\ & [2,f_5,f_6,f_8,3,0,1,f_4,f_{10},4,7],\, [2,f_9,f_5,f_{10},3,0,1,f_6,4,7,f_7], \\ & [2,f_7,f_5,f_8,3,1,f_2,f_3,4,7,f_9],\, [3,f_9,f_4,f_{10},4,f_1,f_5,2,1,f_2,f_6], \\ & [3,f_7,f_4,f_8,6,2,4,f_9,5,7,f_{10}],\, [2,f_3,f_6,4,f_4,f_1,f_{10},1,f_5,0,3], \\ & [f_9,2,f_3,f_{10},4,f_1,0,3,f_2,f_8,1],\, [f_7,2,f_3,f_8,3,f_2,f_5,4,f_1,f_9,1], \\ & [f_4,1,f_3,f_5,2,f_2,f_{10},3,f_9,6,7],\, [1,f_4,f_2,f_7,5,f_1,f_8,3,f_6,2,4], \\ & [f_2,f_9,5,6,f_8,f_4,2,7,f_{10},3,4],\, [1,f_6,f_1,4,f_7,f_8,3,5,f_5,0,2], \\ & [1,3,f_7,5,6,4,7,f_8,f_4,0,2],\, [f_2,2,5,6,f_5,f_3,0,3,f_6,4,7] \bigr \},\\ B'_{5} = \bigl \{& [0,1,f_1,2,3,f_2,4,5,f_3,6,7],\, [0,1,f_2,2,3,f_3,4,5,f_1,6,7], \\ & [0,1,f_3,2,3,f_1,4,5,f_2,6,7],\, [1,2,f_1,3,4,f_2,5,6,f_3,7,0], \\ & [1,2,f_2,3,4,f_3,5,6,f_1,7,0],\, [1,2,f_3,3,4,f_1,5,6,f_2,7,0], \\ & [1,2,0,4,f_1,f_2,3,7,6,f_9,f_{10}],\, [2,3,1,5,f_1,f_2,4,0,7,f_9,f_{10}], \\ & [3,4,2,6,f_1,f_2,5,1,0,f_9,f_{10}],\, [4,5,3,7,f_1,f_2,6,2,1,f_9,f_{10}], \\ & [5,6,4,0,f_3,f_4,7,3,2,f_9,f_{10}],\, [6,7,5,1,f_3,f_4,0,4,3,f_9,f_{10}], \\ & [7,0,6,2,f_3,f_4,1,5,4,f_9,f_{10}],\, [0,1,7,3,f_3,f_4,2,6,5,f_9,f_{10}], \\ & [1,3,4,0,f_5,2,6,f_6,f_1,f_2,7],\, [2,4,5,1,f_5,3,7,f_6,f_1,f_2,0], \\ & [3,5,6,2,f_5,4,0,f_6,f_1,f_2,1],\, [4,6,7,3,f_5,5,1,f_6,f_1,f_2,2], \\ & [5,7,0,4,f_7,6,2,f_8,f_1,f_2,3],\, [6,0,1,5,f_7,7,3,f_8,f_1,f_2,4], \\ & [7,1,2,6,f_7,0,4,f_8,f_1,f_2,5],\, [0,2,3,7,f_7,1,5,f_8,f_1,f_2,6], \\ & [f_9,0,4,f_4,f_5,1,3,5,2,6,7],\, [f_9,1,5,f_4,f_5,2,4,6,3,7,0], \\ & [f_9,2,6,f_4,f_5,3,5,7,4,0,1],\, [f_9,3,7,f_4,f_5,4,6,0,5,1,2], \\ & [f_{10},4,0,f_4,f_5,5,7,1,6,2,3],\, [f_{10},5,1,f_4,f_5,6,0,2,7,3,4], \\ & [f_{10},6,2,f_4,f_5,7,1,3,0,4,5],\, [f_{10},7,3,f_4,f_5,0,2,4,1,5,6]\bigr \}. \end{aligned}$$

Then an \(H_5\)-decomposition of \(K^{(3)}_{18}\setminus K^{(3)}_{10}\) consists of the orbits of the \(H_5\)-blocks in \(B_5\) under the action of the map \(f_i \mapsto f_i\), for \(i \in [1,10]\), and \(j \mapsto j+1 \pmod {8}\) on the vertices along with the \(H_5\)-blocks in \(B'_5\).

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Bunge, R.C., Dawson, E., Donovan, M., Hatzer, C., Maass, J. (2024). On Decompositions of Complete 3-Uniform Hypergraphs into a Linear Forest with 4 Edges. In: Hoffman, F., Holliday, S., Rosen, Z., Shahrokhi, F., Wierman, J. (eds) Combinatorics, Graph Theory and Computing. SEICCGTC 2021. Springer Proceedings in Mathematics & Statistics, vol 448. Springer, Cham. https://doi.org/10.1007/978-3-031-52969-6_29

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