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 Th60:
  G is edgeless iff G.vChromaticNum() = 1
proof
  hereby
    assume A1: G is edgeless;
    then A2: G.vChromaticNum() c= 1 by Th57;
    1 c= G.vChromaticNum() by A1, Th59;
    hence G.vChromaticNum() = 1 by A2, XBOOLE_0:def 10;
  end;
  assume A3: G.vChromaticNum() = 1;
  then G is loopless;
  then G is 1-vcolorable by A3, Th54;
  hence G is edgeless;
end;
