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# 7 Vertex-Colouring and Complete Subgraphs

### Sections

1. VertexColouring
2. CompleteSubgraphs
3. CompleteSubgraphsOfGivenSize

The following sections describe functions for (proper) vertex-colouring or determining complete subgraphs of given graphs. The function `CompleteSubgraphsOfGivenSize` can also be used to determine the complete subgraphs with given vertex-weight sum in a vertex-weighted graphindexvertex-weighted graph, where the weights can be positive integers or non-zero vectors of non-negative integers.

## 7.1 VertexColouring

- `VertexColouring(` gamma`)`

  This function returns a proper vertex-colouring C for the graph gamma, which must be simple.

  This proper vertex-colouring C is a list of positive integers (the colours), indexed by the vertices of gamma, with the property that C [i] ¹ C [j] whenever [i,j] is an edge of gamma. At present a greedy algorithm is used, and the number of colours used is by no means guaranteed to be minimal.

      gap> VertexColouring( JohnsonGraph(4,2) );
      [ 1, 3, 2, 2, 3, 1 ]

  ## 7.2 CompleteSubgraphs

- `CompleteSubgraphs(` gamma`)`
- `CompleteSubgraphs(` gamma`,` k`)`
- `CompleteSubgraphs(` gamma`,` k`,` alls`)`

  Let gamma be a simple graph and k an integer. This function returns a set K of complete subgraphs of gamma, where a complete subgraph is represented by its vertex-set. If k is non-negative then the elements of K each have size k, otherwise the elements of K represent maximal complete subgraphs of gamma. (A maximal complete subgraph of gamma is a complete subgraph of gamma which is not properly contained in another complete subgraph of gamma.) The default for k is -1, i.e. maximal complete subgraphs. See also `CompleteSubgraphsOfGivenSize`, which can be used to compute the maximal complete subgraphs of given size, and can also be used to determine the (maximal or otherwise) complete subgraphs with given vertex-weight sum in a vertex-weighted graph.

  The optional parameter alls controls how many complete subgraphs are returned. The valid values for alls are 0, 1 (the default), and 2.

  Warning: Using the default value of 1 for alls (see below) means that more than one element may be returned for some ``gamma`.group` orbit(s) of the required complete subgraphs. To obtain just one element from each ``gamma`.group` orbit of the required complete subgraphs, you must give the value 2 to the parameter alls.

  If alls=0 (or `false` for backward compatibility) then K will contain at most one element. In this case, if k is negative then K will contain just one maximal complete subgraph, and if k is non-negative then K will contain a complete subgraph of size k if and only if such a subgraph is contained in gamma.

  If alls=1 (or `true` for backward compatibility) then K will contain (perhaps properly) a set of ``gamma`.group` orbit-representatives of the maximal (if k is negative) or size k (if k is non-negative) complete subgraphs of gamma.

  If alls=2 then K will be a set of ``gamma`.group` orbit-representatives of the maximal (if k is negative) or size k (if k is non-negative) complete subgraphs of gamma. This option can be more costly than when alls=1.

  Before applying `CompleteSubgraphs`, one may want to associate the full automorphism group of gamma with gamma, via ``gamma`:= NewGroupGraph( AutGroupGraph(`gamma`),` gamma`);`.

  An alternative name for this function is `Cliques` indexCliques.

  See also CompleteSubgraphsOfGivenSize.

      gap> gamma := JohnsonGraph(5,2);
      rec( isGraph := true, order := 10,
        group := Group([ ( 1, 5, 8,10, 4)( 2, 6, 9, 3, 7), ( 2, 5)( 3, 6)( 4, 7) ]),
        schreierVector := [ -1, 2, 2, 1, 1, 1, 2, 1, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
            [ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ], isSimple := true )
      gap> CompleteSubgraphs(gamma);
      [ [ 1, 2, 3, 4 ], [ 1, 2, 5 ] ]
      gap>  CompleteSubgraphs(gamma,3,2);
      [ [ 1, 2, 3 ], [ 1, 2, 5 ] ]
      gap> CompleteSubgraphs(gamma,-1,0);
      [ [ 1, 2, 5 ] ]

  ## 7.3 CompleteSubgraphsOfGivenSize

- `CompleteSubgraphsOfGivenSize(` gamma`,` k`)`
- `CompleteSubgraphsOfGivenSize(` gamma`,` k`,` alls`)`
- `CompleteSubgraphsOfGivenSize(` gamma`,` k`,` alls`,` maxi`)`
- `CompleteSubgraphsOfGivenSize(` gamma`,` k`,` alls`,` maxi`,` col`)`
- `CompleteSubgraphsOfGivenSize(` gamma`,` k`,` alls`,` maxi`,` col`,` wts`)`

  Let gamma be a simple graph, and k a non-negative integer or vector of non-negative integers. This function returns a set K (possibly empty) of complete subgraphs of size k of gamma. The vertices may have weights, which should be non-zero integers if k is an integer and non-zero d-vectors of non-negative integers if k is a d-vector, and in these cases, a complete subgraph of size k means a complete subgraph whose vertex-weights sum to k. The exact nature of the set K depends on the values of the parameters supplied to this function. A complete subgraph is represented by its vertex-set.

  The optional parameter alls controls how many complete subgraphs are returned. The valid values for alls are 0, 1 (the default), and 2.

  Warning: Using the default value of 1 for alls (see below) means that more than one element may be returned for some ``gamma`.group` orbit(s) of the required complete subgraphs. To obtain just one element from each ``gamma`.group` orbit of the required complete subgraphs, you must give the value 2 to the parameter alls.

  If alls=0 (or `false` for backward compatibility) then K will contain at most one element. If maxi=`false` then K will contain one element if and only if gamma contains a complete subgraph of size k. If maxi=`true` then K will contain one element if and only if gamma contains a maximal complete subgraph of size k, in which case K will contain (the vertex-set of) such a maximal complete subgraph. (A maximal complete subgraph of gamma is a complete subgraph of gamma which is not properly contained in another complete subgraph of gamma.)

  If alls=1 (or `true` for backward compatibility) and maxi=`false`, then K will contain (perhaps properly) a set of ``gamma`.group` orbit-representatives of the size k complete subgraphs of gamma. If alls=1 (the default) and maxi=`true`, then K will contain (perhaps properly) a set of ``gamma`.group` orbit-representatives of the size k maximal complete subgraphs of gamma.

  If alls=2 and maxi=`false`, then K will be a set of ``gamma`.group` orbit-representatives of the size k complete subgraphs of gamma. If alls=2 and maxi=`true` then K will be a set of ``gamma`.group` orbit-representatives of the size k maximal complete subgraphs of gamma. This option can be more costly than when alls=1.

  The optional parameter maxi controls whether only maximal complete subgraphs of size k are returned. The default is `false`, which means that non-maximal as well as maximal complete subgraphs of size k are returned. If maxi=`true` then only maximal complete subgraphs of size k are returned. (Previous to version 4.1 of GRAPE, maxi=`true` meant that it was assumed (but not checked) that all complete subgraphs of size k were maximal.)

  The optional boolean parameter col is used to determine whether or not partial proper vertex-colouring is used to cut down the search tree. The default is `true`, which says to use this partial colouring. For backward compatibility, col a rational number means the same as col=`true`.

  The optional parameter wts should be a list of vertex-weights; the list should be of length ``gamma`.order`, with the i-th element being the weight of vertex i. The weights must be all positive integers if k is an integer, and all non-zero d-vectors of non-negative integers if k is a d-vector. The default is that all weights are equal to 1. (Recall that a complete subgraph of size k means a complete subgraph whose vertex-weights sum to k.)

  If wts is a list of integers, then this list must be ``gamma`.group` invariant, where the action permutes the list positions in the natural way.

  If wts is a list of d-vectors then we assume that ``gamma`.group` acts on the set of all integer d-vectors by permuting vector positions, such that, for all v in `[1..`gamma`.order]` and all g in ``gamma`.group`, we have wts [vg] = wts [v]g (where the first action is `OnPoints` and for the second action, if ig=j then (wts [v]g)[j]=wts [v][i]), and that we also have k g=k . These assumptions are not checked by the function, and the use of vector-weights is primarily for advanced users of GRAPE.

  An alternative name for this function is `CliquesOfGivenSize` indexCliquesOfGivenSize.

  See also CompleteSubgraphs.

      gap> gamma:=JohnsonGraph(6,2);
      rec( isGraph := true, order := 15,
        group := Group([ ( 1, 6,10,13,15, 5)( 2, 7,11,14, 4, 9)( 3, 8,12),
            ( 2, 6)( 3, 7)( 4, 8)( 5, 9) ]),
        schreierVector := [ -1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 6, 7, 8, 9 ] ], representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 2, 3 ],
            [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 5 ],
            [ 4, 6 ], [ 5, 6 ] ], isSimple := true )
      gap> CompleteSubgraphsOfGivenSize(gamma,4);
      [ [ 1, 2, 3, 4 ] ]
      gap> CompleteSubgraphsOfGivenSize(gamma,4,1,true);
      [  ]
      gap> CompleteSubgraphsOfGivenSize(gamma,5,2,true);
      [ [ 1, 2, 3, 4, 5 ] ]
      gap> delta:=NewGroupGraph(Group(()),gamma);;
      gap> CompleteSubgraphsOfGivenSize(delta,5,2,true);
      [ [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 2, 6, 10, 11, 12 ],
        [ 3, 7, 10, 13, 14 ], [ 4, 8, 11, 13, 15 ], [ 5, 9, 12, 14, 15 ] ]
      gap> CompleteSubgraphsOfGivenSize(delta,5,0);
      [ [ 1, 2, 3, 4, 5 ] ]
      gap> CompleteSubgraphsOfGivenSize(delta,5,1,false,true,
      >       [1,2,3,4,5,6,7,8,7,6,5,4,3,2,1]);
      [ [ 1, 4 ], [ 2, 3 ], [ 3, 14 ], [ 4, 15 ], [ 5 ], [ 11 ], [ 12, 15 ],
        [ 13, 14 ] ]

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