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 Th165:
  G is edgeless iff G is 1-tcolorable
proof
  hereby
    assume A1: G is edgeless;
    then G is 1-vcolorable;
    then consider f being VColoring of G such that
      A2: f is proper & card rng f c= 1;
    reconsider t = [f,{}] as TColoring of G by A1, Th137;
    A3: t is proper by A2;
    card((rng t_V)\/rng t_E) = card((rng t_V)\/rng {})
      .= card rng t_V;
    hence G is 1-tcolorable by A2, A3;
  end;
  assume G is 1-tcolorable;
  then G is 1-vcolorable by Th162;
  hence thesis;
end;
