oper: add IsDigraphColouring

A method that verifies if a list or transformation is a colouring of a digraph.

doc/grahom.xml

@@ -541,3 +541,47 @@ false]]></Example>
   </Description>
 </ManSection>
 <#/GAPDoc>
+
+<#GAPDoc Label="IsDigraphColouring">
+  <ManSection>
+    <Oper Name="IsDigraphColouring" Arg="digraph, list"/>
+    <Oper Name="IsDigraphColouring" Arg="digraph, t" 
+      Label="for a transformation"/>
+    <Returns> <K>true</K> or <K>false</K>. </Returns>
+    <Description>
+      The operation <C>IsDigraphColouring</C> verifies whether or not 
+      the list <A>list</A> describes a proper colouring of the digraph 
+      <A>digraph</A>. 
+      <P/>
+
+      A list <A>list</A> describes a <E>proper colouring</E> of a digraph
+      <A>digraph</A> if <A>list</A> consists of positive integers, the length of
+      <A>list</A> equals the number of vertices in <A>digraph</A>, and for any
+      vertices <C>u, v</C> of <A>digraph</A> if <C>u</C> and <C>v</C> are adjacent,
+      then <C><A>list</A>[u] &gt;&lt; <A>list</A>[v]</C>.
+      <P/>
+
+      A transformation <A>t</A> describes a proper colouring of a digraph
+      <A>digraph</A>, if <C>ImageListOfTransformation(<A>t</A>,
+      DigraphNrVertices(<A>digraph</A>))</C> is a proper colouring of 
+      <A>digraph</A>. <P/>
+
+      See also <Ref Oper="IsDigraphHomomorphism"/>.
+
+      <Example><![CDATA[
+gap> D := JohnsonDigraph(5, 3);
+<digraph with 10 vertices, 60 edges>
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);
+true
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);
+false
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);
+false
+gap> IsDigraphColouring(D, [1, 2, 3]);
+false
+gap> IsDigraphColouring(D, IdentityTransformation);
+true
+]]></Example>
+    </Description>
+  </ManSection>
+<#/GAPDoc>

doc/z-chap6.xml

@@ -40,6 +40,7 @@ from} $E_a$ \emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are
     <#Include Label="IsomorphismDigraphsColours">
     <#Include Label="RepresentativeOutNeighbours">
     <#Include Label="IsDigraphAutomorphism">
+    <#Include Label="IsDigraphColouring">
   </Section>
 
   <Section><Heading>Homomorphisms of digraphs</Heading>

gap/grahom.gd

@@ -54,3 +54,6 @@ DeclareOperation("IsDigraphMonomorphism",
                  [IsDigraph, IsDigraph, IsPerm]);
 DeclareOperation("IsDigraphEmbedding",
                  [IsDigraph, IsDigraph, IsPerm]);
+
+DeclareOperation("IsDigraphColouring", [IsDigraph, IsList]);
+DeclareOperation("IsDigraphColouring", [IsDigraph, IsTransformation]);

gap/grahom.gi

@@ -549,3 +549,31 @@ function(src, ran, x)
   od;
   return true;
 end);
+
+InstallMethod(IsDigraphColouring, "for a digraph and a list",
+[IsDigraph, IsHomogeneousList],
+function(digraph, colours)
+  local n, out, v, w;
+  n := DigraphNrVertices(digraph);
+  if Length(colours) <> n or ForAny(colours, x -> not IsPosInt(x)) then
+    return false;
+  fi;
+  out := OutNeighbours(digraph);
+  for v in DigraphVertices(digraph) do
+    for w in out[v] do
+      if colours[w] = colours[v] then
+        return false;
+      fi;
+    od;
+  od;
+  return true;
+end);
+
+InstallMethod(IsDigraphColouring, "for a digraph and a transformation",
+[IsDigraph, IsTransformation],
+function(digraph, t)
+  local n;
+  n := DigraphNrVertices(digraph);
+  return IsDigraphColouring(digraph,
+                            ImageListOfTransformation(t, n));
+end);

tst/standard/grahom.tst

@@ -1084,6 +1084,20 @@ true
 gap> IsDigraphEmbedding(src, ran, ());
 false
 
+# IsDigraphColouring
+gap> D := JohnsonDigraph(5, 3);
+<digraph with 10 vertices, 60 edges>
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);
+true
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);
+false
+gap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);
+false
+gap> IsDigraphColouring(D, [1, 2, 3]);
+false
+gap> IsDigraphColouring(D, IdentityTransformation);
+true
+
 #T# DIGRAPHS_UnbindVariables
 gap> Unbind(edges);
 gap> Unbind(epis);