reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th37:
  for G2 being non edgeless _Graph, v,e,w being object
  for G1 being addAdjVertex of G2,v,e,w
  holds G1 is c-vcolorable iff G2 is c-vcolorable
proof
  let G2 be non edgeless _Graph, v,e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  hereby
    assume A1: G1 is c-vcolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is c-vcolorable by A1, Th31;
  end;
  assume A2: G2 is c-vcolorable;
  per cases;
  suppose not e in the_Edges_of G2 & v in the_Vertices_of G2 &
      not w in the_Vertices_of G2;
    hence thesis by A2, Lm4;
  end;
  suppose A3: not e in the_Edges_of G2 & not v in the_Vertices_of G2 &
      w in the_Vertices_of G2;
    set G3 = the reverseEdgeDirections of G1, {e};
    G3 is addAdjVertex of G2,w,e,v by A3, GLIBPRE1:66;
    then G3 is c-vcolorable by A2, A3, Lm4;
    hence thesis by Th33;
  end;
  suppose not((not e in the_Edges_of G2 & v in the_Vertices_of G2 &
      not w in the_Vertices_of G2) or (not e in the_Edges_of G2 &
      not v in the_Vertices_of G2 & w in the_Vertices_of G2));
    then G1 == G2 by GLIB_006:def 12;
    hence thesis by A2, Th32;
  end;
end;
