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7 Vertex-Colouring and Complete Subgraphs

Sections

  1. VertexColouring
  2. MinimumVertexColouring
  3. ChromaticNumber
  4. CompleteSubgraphs
  5. CompleteSubgraphsOfGivenSize
  6. MaximumClique
  7. CliqueNumber

The following sections describe functions for (proper) vertex-colouring and determining complete subgraphs of a given simple graph. Included are functions for determining the chromatic number and the clique number of a simple graph.

The function CompleteSubgraphsOfGivenSize can be used to determine the complete subgraphs with given vertex-weight sum in a vertex-weighted graph indexvertex-weighted graph, where the weights can be positive integers or non-zero vectors of non-negative integers.

7.1 VertexColouring

  • VertexColouring( gamma )
  • VertexColouring( gamma, k )
  • VertexColouring( gamma, k, m )

    This function returns a proper vertex-colouring C for the graph gamma, which must be simple. A proper vertex-colouring indexproper vertex-colouring of gamma is an assignment of colours to the vertices of gamma, such that, if [i,j] is an edge, then vertices i and j are assigned different colours.

    The returned proper vertex-colouring C is given as a list of positive integers (the colours), indexed by the vertices of gamma, with the property that C [i] ≠ C [j] whenever [i,j] is an edge of gamma.

    If the optional parameter k is given, then k must be a non-negative integer. In this case, a proper vertex-colouring using at most k colours is returned, if such a colouring exists, and fail otherwise.

    If, in addition to k, the optional parameter m is given, then m must be a a non-negative integer, such that there is no monochromatic set of vertices of size greater than m in any proper vertex-colouring of gamma which uses at most k colours. This information (which is not checked) may help to speed up the function.

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

    7.2 MinimumVertexColouring

  • MinimumVertexColouring( gamma )

    This function returns a minimum vertex-colouring C for the graph gamma, which must be simple. A minimum vertex-colouring indexminimum vertex-colouring is a proper vertex-colouring using as few colours as possible.

    The returned minimum vertex-colouring C is given as a list of positive integers (the colours), indexed by the vertices of gamma, with the property that C [i] ≠ C [j] whenever [i,j] is an edge of gamma, and subject to this property, the number of distinct elements of C is as small as possible.

    See also VertexColouring.

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

    7.3 ChromaticNumber

  • ChromaticNumber( gamma )

    This function returns the chromatic number of the given graph gamma, which must be simple. The chromatic number indexchromatic number of gamma is the minimum number of colours needed to properly vertex-colour gamma, that is, the number of colours used in a minimum vertex-colouring of gamma.

    See also MinimumVertexColouring.

    gap> ChromaticNumber(JohnsonGraph(5,2));
    5
    gap> ChromaticNumber(JohnsonGraph(6,2));
    5
    gap> ChromaticNumber(JohnsonGraph(7,2));
    7
    

    7.4 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 ] ]
    

    7.5 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 ] ]
    

    7.6 MaximumClique

  • MaximumClique( gamma )

    This function returns a maximum clique of the graph gamma, which must be simple. A maximum clique indexmaximum clique of gamma is a set of pairwise adjacent vertices of gamma of the largest possible size.

    An alternative name for this function is MaximumCompleteSubgraph indexMaximumCompleteSubgraph.

    See also CompleteSubgraphsOfGivenSize.

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

    7.7 CliqueNumber

  • CliqueNumber( gamma )

    This function returns the clique number of the given graph gamma, which must be simple. The clique number indexclique number of gamma is the size of a largest clique in gamma, where a clique is a set of pairwise adjacent vertices.

    gap> CliqueNumber(JohnsonGraph(5,2));
    4
    gap> CliqueNumber(JohnsonGraph(6,2));
    5
    gap> CliqueNumber(JohnsonGraph(7,2));
    6
    

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    grape manual
    June 2018