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

theorem Th172:
  for G2 being non edgeless _Graph, v,e,w being object
  for G1 being addAdjVertex of G2,v,e,w
  st G2 is c-tcolorable holds G1 is (c+`1)-tcolorable
proof
  let G2 be non edgeless _Graph, v,e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  assume A1: G2 is c-tcolorable;
  per cases;
  suppose not e in the_Edges_of G2 & not v in the_Vertices_of G2
      & w in the_Vertices_of G2;
    hence thesis by A1, Lm18;
  end;
  suppose A2: not e in the_Edges_of G2 & v in the_Vertices_of G2
      & not w in the_Vertices_of G2;
    set G3 = the reverseEdgeDirections of G1, {e};
    G3 is addAdjVertex of G2,w,e,v by A2, GLIBPRE1:66;
    then G3 is (c+`1)-tcolorable by A1, A2, Lm18;
    hence thesis by Th168;
  end;
  suppose not((not e in the_Edges_of G2 & not v in the_Vertices_of G2
      & w in the_Vertices_of G2)or(not e in the_Edges_of G2
      & v in the_Vertices_of G2 & not w in the_Vertices_of G2));
    then G1 == G2 by GLIB_006:def 12;
    then G1 is c-tcolorable by A1, Th167;
    hence thesis by Th161, CARD_2:94;
  end;
end;
