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Question: Is it possible to decompose $K_{12,12}$ into four edge-disjoint copies of $3(K_{4,4}-I)$, where $I$ denotes a $1$-factor?

Here's a drawing of $3(K_{4,4}-I)$:

drawing of the graph

The same motivation for my question Are there $3$ disjoint copies of $2K_{3,3} \cup (K_{5,5} \setminus C_{10})$ in $K_{11,11}$? but just another special case.

  • The number of edges in $3(K_{4,4}-I)$ is $36$ and the number of edges in $K_{12,12}$ is $144 = 4 \times 36$.

  • The graph $3(K_{4,4}-I)$ is $3$-regular, and $K_{12,12}$ is $(4 \times 3)$-regular.

I previously asked Does $K_{12,12}$ decompose into $K_{4,4}-I$ subgraphs? which shows that $K_{12,12}$ decomposes into $12$ edge-disjoint copies of $K_{4,4}-I$, which is a necessary condition for the decomposition in this question.

1 Answers 1

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Here is one such decomposition, found by simulated annealing:

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Here are the details of the graphs used:

  • The second graph matches vertices $4,9,10,11$ to $8,5,6,4$, vertices $3,5,6,7$ to $12,3,10,9$, and vertices $1,2,8,12$ to $2,1,7,11$.
  • The third graph matches vertices $5,7,8,9$ to $9,3,1,4$, vertices $1,2,3,12$ to $12,8,5,6$, and vertices $4,6,10,11$ to $2,7,11,10$.
  • The fourth graph matches vertices $1,2,4,8$ to $8,12,10,9$, vertices $3,5,6,9$ to $2,7,5,11$, and vertices $6,10,11,12$ to $3,4,6,1$.

The actual answer may not be as interesting nearly two years later, but here is my Mathematica code for the simulated annealing, which can be more broadly useful.

(Here, each $3K_{4,4}-I$ is represented by a pair of permutations, one of the top and one of the bottom vertices. Each step we take is randomly switching two of the numbers in one of the 8 permutations we have. The energy value of a state is the number of edges of $K_{12,12}$ not covered, which we eventually bring down to $0$.)

edges[{perm1_, perm2_}] :=
  Join[
    Tuples[{perm1[[1 ;; 4]], perm2[[1 ;; 4]]}],
    Tuples[{perm1[[5 ;; 8]], perm2[[5 ;; 8]]}],
    Tuples[{perm1[[9 ;; 12]], perm2[[9 ;; 12]]}]]~Complement~
   Table[{perm1[[i]], perm2[[i]]}, {i, 1, 12}];

value[state_] := 12^2 - Length[Union @@ (edges /@ state)];
randomPerm[] := {RandomSample[Range[12]], RandomSample[Range[12]]}
randomSwitch[state_] := 
 Module[{h = RandomInteger[{1, 4}], i = RandomInteger[{1, 2}], j, k, 
   copy = state},
  {j, k} = RandomSample[Range[12], 2];
  copy[[h, i, {j, k}]] = Reverse[copy[[h, i, {j, k}]]];
  Return[copy];
  ]

currentState = 
  bestState = {randomPerm[], randomPerm[], randomPerm[], randomPerm[]};
currentEnergy = bestEnergy = value[currentState];
temp = 1;
While[Exp[-1/temp] > 1/1000,
 Do[
  nextState = randomSwitch[currentState];
  nextEnergy = value[nextState];
  If[nextEnergy < bestEnergy, bestState = nextState; 
   bestEnergy = nextEnergy];
  prob = Exp[-((nextEnergy - currentEnergy)/temp)];
  If[RandomReal[] < prob, currentState = nextState; 
   currentEnergy = nextEnergy];
  , {2000}];
 If[bestEnergy == 0, Break[]];
 temp *= 0.99; Print[{temp, currentEnergy}]
 ]
Print["Done ", bestEnergy];