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 Th38:
  for G2 being edgeless _Graph, v,e,w being object
  for G1 being addAdjVertex of G2,v,e,w holds G1 is 2-vcolorable
proof
  let G2 be edgeless _Graph, v,e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  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 Lm5;
  end;
  suppose A1: 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 A1, GLIBPRE1:66;
    then G3 is 2-vcolorable by A1, Lm5;
    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 & G2 is 1-vcolorable by GLIB_006:def 12;
    then G1 is 1-vcolorable by Th32;
    hence thesis by Th27, FIELD_5:3;
  end;
end;
