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 Th196:
  for G being loopless non edgeless _Graph holds 3 c= G.tChromaticNum()
proof
  let G be loopless non edgeless _Graph;
  assume not 3 c= G.tChromaticNum();
  then G.tChromaticNum() in 3 by ORDINAL1:16;
  then per cases by CARD_1:51, ENUMSET1:def 1;
  suppose G.tChromaticNum() = 0;
    hence contradiction;
  end;
  suppose G.tChromaticNum() = 1;
    hence contradiction by Th195;
  end;
  suppose G.tChromaticNum() = 2;
    then G is 2-tcolorable by Th186;
    then consider t being TColoring of G such that
      A1: t is proper & card((rng t_V)\/rng t_E) c= 2;
    2 c= card((rng t_V)\/rng t_E)
    proof
      assume not 2 c= card((rng t_V)\/rng t_E);
      then card((rng t_V)\/rng t_E) in 2 by ORDINAL1:16;
      then per cases by CARD_1:50, TARSKI:def 2;
      suppose card((rng t_V)\/rng t_E) = 0;
        hence contradiction;
      end;
      suppose card((rng t_V)\/rng t_E) = 1;
        then G is 1-tcolorable by A1;
        hence contradiction;
      end;
    end;
    then consider a,b being object such that
      A2: a<>b & (rng t_V)\/rng t_E = {a,b} by A1, XBOOLE_0:def 10, CARD_2:60;
    set e = the Element of the_Edges_of G;
    set v = (the_Source_of G).e, w = (the_Target_of G).e;
    e Joins v,w,G & e Joins w,v,G by GLIB_000:def 13;
    then A3: t_V.v <> t_V.w & t_V.v <> t_E.e & t_V.w <> t_E.e by A1, Th148;
    e in the_Edges_of G;
    then e in dom t_E by PARTFUN1:def 2;
    then t_E.e in rng t_E by FUNCT_1:3;
    then t_E.e in {a,b} by A2, XBOOLE_0:def 3;
    then A4: t_E.e = a or t_E.e = b by TARSKI:def 2;
    v in the_Vertices_of G & w in the_Vertices_of G;
    then v in dom t_V & w in dom t_V by PARTFUN1:def 2;
    then t_V.v in rng t_V & t_V.w in rng t_V by FUNCT_1:3;
    then t_V.v in {a,b} & t_V.w in {a,b} by A2, XBOOLE_0:def 3;
    then (t_V.v = a or t_V.v = b) & (t_V.w = a or t_V.w = b) by TARSKI:def 2;
    hence contradiction by A3, A4;
  end;
end;
