string GeneralizeBoards(List <SuperSlimBoard> boards)
{
if (UseWildCards)
{
var examples = boards.Select(b => ToListIndices(b)).ToList();
var nonExamples = Enumerable.Repeat(_possibleListIndices, _activeIndices.Count).CartesianProduct().Select(ll => ll.ToList()).Except(examples.Distinct(_sequenceComparer), _sequenceComparer).ToList();
var generalized = _sequenceGeneralizer.Generalize(examples, nonExamples, false);
return(generalized.Select(gg => "$" + string.Join("|", gg.Select((_, i) => _.ToTex(_possibleLists, _activeListSizes[i]))) + "$").Listify("and"));
}
return(boards.Select(b => ToListString(b)).OrderBy(x => x).Select(s => "$" + s + "$").Listify("and"));
}
public override string WriteProof()
{
var length = Mind.ColorableBoards[0].ToXYZ().Length;
var allBoards = Mind.ColorableBoards.Union(Mind.NonColorableBoards).ToList();
var comparer = new SequenceGeneralizer <int> .VectorComparer();
var sg = new SequenceGeneralizer <int>(length, new List <int> {
0, 1, 2
});
var zot2 = allBoards.Select(b => b.To012()).ToList();
var examples2 = allBoards.Select(b => b.To012()).ToList();
var nonExamples2 = Enumerable.Repeat(Enumerable.Range(0, 3), length).CartesianProduct().Select(ll => ll.ToList()).Except(zot2.Distinct(comparer), comparer).ToList();
var generalized2 = sg.Generalize(examples2, nonExamples2, false);
var allBoardsXYZ = generalized2.Select(gg => "$" + string.Join("", gg.Select(_ => _.ToTex())) + "$").Listify("or");
var sb = new StringBuilder();
var figureID = "fig:" + Guid.NewGuid().ToString();
sb.AppendLine("\\begin{figure}");
sb.AppendLine("\\centering");
sb.AppendLine(_figureTikz);
sb.AppendLine("\\caption{Solid vertices have lists of size 3 and the labeled vertices have lists of size 2.}\\label{" + figureID + "}");
sb.AppendLine("\\end{figure}");
sb.AppendLine("\\begin{lem}");
sb.AppendLine("The graph in Figure \\ref{" + figureID + "} is reducible.");
sb.AppendLine("\\end{lem}");
var letters = new List <string>()
{
"X", "Y", "Z"
};
var stringLength = Mind.ColorableBoards[0].Stacks.Value.Count(ss => ss.PopulationCount() == 2);
var rng = new Random(DateTime.Now.Millisecond);
var randomString = "";
for (int i = 0; i < stringLength; i++)
{
randomString += letters[rng.Next(3)];
}
var randomString2 = "";
for (int i = 0; i < stringLength; i++)
{
randomString2 += letters[rng.Next(3)];
}
var randomString3 = "";
for (int i = 0; i < stringLength; i++)
{
randomString3 += letters[rng.Next(3)];
}
sb.Append("\\begin{proof}");
sb.AppendLine("Let $X = \\{0,1\\}$, $Y = \\{0,2\\}$ and $Z = \\{1,2\\}$. Then with the vertex ordering in Figure \\ref{" + figureID + "}, a string such as " + randomString + ", ");
sb.AppendLine("represents a possible list assignment on $V(H)$ arising from a $3$-edge-coloring of $G-E(H)$.");
sb.AppendLine("By an $X$-Kempe change, we mean flipping colors $0$ and $1$ on a two-colored path in $G-E(H)$. We call such a path an $X$-path. ");
sb.AppendLine("Any endpoint of an $X$-path in $H$ must end at a $Y$ or $Z$ vertex. The meanings of $Y$-Kempe change, $Z$-Kempe change, $Y$-path and $Z$-path are analogous.");
sb.AppendLine("Note that if there are an odd number of $Y$'s and $Z$'s, then at least one $X$-path has only one endpoint in $H$.");
sb.AppendLine("We use shorthand notation like $\\K_{X, 2}(" + randomString + ",5,6) \\Rightarrow " + randomString2 + "," + randomString3 + "$ (Case 1).");
sb.AppendLine("This means the $X$-Kempe change on " + randomString + " starting at the second vertex and ending at the fifth and sixth result in boards " + randomString2 + " and " + randomString3 + " respectively and these are handled by Case 1.");
sb.AppendLine("The $\\infty$ symbol means starting (or ending) outside $H$.");
sb.AppendLine();
if (Mind.OnlyConsiderNearlyColorableBoards)
{
sb.AppendLine("We need to handle all boards that are nearly colorable for edge $e$ up to permutations of $\\{X,Y,Z\\}$, so it will suffice to handle all boards of the form " + allBoardsXYZ + ".");
}
else
{
sb.AppendLine("We need to handle all boards up to permutations of $\\{X,Y,Z\\}$, so it will suffice to handle all boards of the form " + allBoardsXYZ + ".");
}
sb.AppendLine();
var wonBoards = new List <SuperSlimBoard>();
for (int caseNumber = 1; caseNumber <= Cases.Count; caseNumber++)
{
var c = Cases[caseNumber - 1];
var boards = c.Boards;
List <SuperSlimBoard> thisClaimBoards;
if (caseNumber > 1)
{
thisClaimBoards = boards.SelectMany(b => new[] { b }.Union(_permutationLinked[b].Select(tup => tup.Item2))).ToList();
}
else
{
thisClaimBoards = boards;
}
wonBoards.AddRange(thisClaimBoards);
var zot = thisClaimBoards.Select(b => b.To012()).ToList();
var examples = thisClaimBoards.Select(b => b.To012()).ToList();
var nonExamples = Enumerable.Repeat(Enumerable.Range(0, 3), length).CartesianProduct().Select(ll => ll.ToList()).Except(zot.Distinct(comparer), comparer).ToList();
var generalized = sg.Generalize(examples, nonExamples);
var boardsXYZ = generalized.Select(gg => "$" + string.Join("", gg.Select(_ => _.ToTex())) + "$").Listify("or");
sb.AppendLine();
sb.AppendLine("\\bigskip");
sb.AppendLine(string.Format("\\case{{{0}}}{{$B$ is one of " + boardsXYZ + ".}}", caseNumber));
sb.AppendLine();
sb.AppendLine("\\bigskip");
if (caseNumber == 1)
{
sb.AppendLine();
sb.AppendLine("In all these cases, $H$ is immediately colorable from the lists.");
}
else
{
sb.AppendLine();
var fixGroups = boards.GroupBy(b =>
{
var treeInfo = Mind.GetWinTreeInfo(b);
var fixLetter = ((long)((1 << treeInfo.First().Alpha) | (1 << treeInfo.First().Beta))).ToXYZ();
return(fixLetter);
});
foreach (var fixGroup in fixGroups)
{
var others = letters.ToList();
others.Remove(fixGroup.Key);
var swapCountGroups = fixGroup.GroupBy(b => Mind.GetWinTreeInfo(b).Max(ti => ti.SwapVertices.Count)).ToList();
foreach (var swapCountGroup in swapCountGroups)
{
if (swapCountGroup.Key == 1)
{
foreach (var b in swapCountGroup)
{
var treeInfo = Mind.GetWinTreeInfo(b);
var groups = treeInfo.GroupBy(ss => ss.SwapVertices[0]);
sb.Append("$\\K_{" + fixGroup.Key + ",\\infty}(" + b.ToXYZ() + "," + groups.OrderBy(gg => gg.Key).Select(gg => gg.Key.GetXYZIndex(b) + 1).Listify(null) + ")");
sb.AppendLine("\\Rightarrow $" + groups.OrderBy(gg => gg.Key).Select(gg => "$" + GetChildBoardName(b, gg.First()) + "$").Listify(null) + "( Case " + treeInfo.Select(bc => GetHandledCaseNumber(b, bc)).Distinct().OrderBy(xx => xx).Listify() + ").");
sb.AppendLine();
if (_permutationLinked[b].Count > 0)
{
sb.AppendLine();
sb.AppendLine();
sb.AppendLine(_permutationLinked[b].Select(ppp => "$" + ppp.Item1 + "\\Rightarrow " + ppp.Item2.ToXYZ() + "$").Listify());
sb.AppendLine();
sb.AppendLine();
}
}
}
else if (swapCountGroup.Key == 2)
{
foreach (var b in swapCountGroup)
{
var treeInfo = Mind.GetWinTreeInfo(b);
var leftover = treeInfo.ToList();
while (leftover.Count > 0)
{
var commonestSwapper = Enumerable.Range(0, b._stackCount).MaxIndex(v => leftover.Count(bc => bc.SwapVertices.Contains(v)));
var handledAll = leftover.Where(bc => bc.SwapVertices.Contains(commonestSwapper)).ToList();
var handled = handledAll.Distinct(bc => bc.SwapVertices.Count == 1 ? -1 : bc.SwapVertices.Except(commonestSwapper).First()).ToList();
sb.Append("$\\K_{" + fixGroup.Key + "," + (commonestSwapper.GetXYZIndex(b) + 1) + "}(" + b.ToXYZ());
var single = handled.FirstOrDefault(bc => bc.SwapVertices.Count == 1);
if (single != null)
{
sb.Append(",\\infty");
}
if (handled.Where(bc => bc.SwapVertices.Count > 1).Count() > 0)
{
sb.Append("," + handled.Where(bc => bc.SwapVertices.Count > 1).OrderBy(bc => bc.SwapVertices.Except(commonestSwapper).First()).Select(bc => bc.SwapVertices.Except(commonestSwapper).First().GetXYZIndex(b) + 1).Listify(null));
}
sb.Append(")");
sb.AppendLine("\\Rightarrow $" + handled.OrderBy(bc => bc.SwapVertices.Count == 1 ? -1 : bc.SwapVertices.Except(commonestSwapper).First()).Select(bc => "$" + GetChildBoardName(b, bc) + "$").Listify(null) + "(Case " + handled.Select(bc => GetHandledCaseNumber(b, bc)).Distinct().OrderBy(xx => xx).Listify() + ").");
sb.AppendLine();
foreach (var bc in handledAll)
{
leftover.Remove(bc);
}
}
if (_permutationLinked[b].Count > 0)
{
sb.AppendLine();
sb.AppendLine();
sb.AppendLine(_permutationLinked[b].Select(ppp => "$" + ppp.Item1 + "\\Rightarrow " + ppp.Item2.ToXYZ() + "$").Listify());
sb.AppendLine();
sb.AppendLine();
}
}
}
}
}
}
sb.AppendLine();
}
sb.AppendLine("\\end{proof}");
return(sb.ToString());
}
