<< DiscreteMath`Combinatorica`

coxeterverts  =  {{1, 2, 3}, {4, 5, 7}, {1, 2, 6}, {3, 4, 5}, {1, 
            2, 7}, {3, 4, 6}, {1, 5, 7}, {2, 3, 4}, {1, 5, 6}, {2, 3, 7}, {1, 
            4, 5}, {2, 6, 7}, {1, 3, 4}, {2, 5, 6}, {1, 3, 7}, {2, 4, 6}, {3, 
            5, 7}, {1, 4, 6}, {2, 3, 5}, {1, 4, 7}, {3, 5, 6}, {2, 4, 7}, {1, 
            3, 6}, {2, 4, 5}, {3, 6, 7}, {1, 2, 5}, {4, 6, 7}, {5, 6, 
            7}};

o4verts  =  KSubsets[{1, 2, 3, 4, 5, 6, 7}, 3];

Complement[o4verts, coxeterverts]

ShowGraph[
        FromUnorderedPairs[
          Union[Partition[
              Flatten[Table[
                  If[Length[
                        Intersection[coxeterverts[[a]], 
                          coxeterverts[[b]]]] == 0, 
                    Sort[{a, b}], {}], {a, 1, 28}, {b, 1, 28}]], 
              2]]]];

kk  =  
        Graph[{{{1, 2}}, {{1, 27}}, {{1, 28}}, {{2, 3}}, {{2, 23}}, {{3, 
                4}}, {{3, 17}}, {{4, 5}}, {{4, 12}}, {{5, 6}}, {{5, 
                21}}, {{6, 7}}, {{6, 26}}, {{7, 8}}, {{7, 16}}, {{8, 
                9}}, {{8, 28}}, {{9, 10}}, {{9, 22}}, {{10, 11}}, {{10, 
                18}}, {{11, 12}}, {{11, 25}}, {{12, 13}}, {{13, 14}}, {{13, 
                28}}, {{14, 15}}, {{14, 20}}, {{15, 16}}, {{15, 24}}, {{16, 
                17}}, {{17, 18}}, {{18, 19}}, {{19, 20}}, {{19, 27}}, {{20, 
                21}}, {{21, 22}}, {{22, 23}}, {{23, 24}}, {{24, 25}}, {{25, 
                26}}, {{26, 27}}}, {{{-1.7313433, 
                0.46875}}, {{-0.70895517, 0.578125}}, {{-1.1791046, 
                0.15625}}, {{-0.9029851, 
                0.0703125}}, {{-0.619403, -0.3125}}, {{-0.8134328, -0.453125}}, {{-1.0447762, -0.3515625}}, {{-0.03731346, 
                0.390625}}, {{0.34328365, -0.375}}, {{-0.029850721, -0.84375}}, {{-0.16417909, -0.3828125}}, {{-0.75373137, 
                0.28125}}, {{-1.4253731, 
                0.3671875}}, {{-1.5597014, -1.1796875}}, {{-0.9776119, -1.0859375}}, {{-1.0671642, -0.8984375}}, {{-1.3059702, -1.015625}}, {{-0.9104477, -1.3125}}, {{-1.7313433, -1.4296875}}, {{-0.9776119, -1.6484375}}, {{-0.37313437, -1.3828125}}, {{0.18656707, -1.0078125}}, {{0.03731346, -0.40625}}, {{-0.21641791, -0.7265625}}, {{-0.36567163, -0.5859375}}, {{-0.79104483, -0.6796875}}, {{-1.8880597, -0.6796875}}, {{-0.61194026, 
                0.859375}}}];

ShowGraph[kk,  EdgeColor ->  Yellow, 
        VertexLabel ->  coxeterverts];

?Vert*

o4verts  =  KSubsets[{1, 2, 3, 4, 5, 6, 7}, 3];

Complement[o4verts, coxeterverts]

ShowGraph[
        FromUnorderedPairs[
          Union[Partition[
              Flatten[Table[
                  If[Length[
                        Intersection[o4verts[[a]], 
                          o4verts[[b]]]] == 0, 
                    Sort[{a, b}], {}], {a, 1, 35}, {b, 1, 35}]], 
              2]]]];

InputForm[
      FromUnorderedPairs[
        Union[Partition[
            Flatten[Table[
                If[Length[
                      Intersection[o4verts[[a]], 
                        o4verts[[b]]]] == 0, Sort[{a, b}], {}], {a, 
                  1, 35}, {b, 1, 35}]], 2]]]]