By Percy Alexander MacMahon
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Additional resources for Combinatory analysis
6 GROUPS, AND POLYNOMIALS OF GRAPHS Five groups are associated with every nonempty graph G: the degree preserving group r d(G), the (vertex) group r(G), the induced edge group r,(G), the edge group r'(G), and the total group r"(G). The total group ofG is the group of all association preserving permutations ofthe elements of G. For the isomorphism of any two of these five groups necessary and sufficient conditions for G exist. For example, we have the following theorem . 1 For any graph G 1= KI, r(G) ~ r"(G) if and only if no component of G is either a cycle or a complete graph.
38] C. J. H. McDiarmid and B. Reed, On total colourings of graphs, J. Comb. Theory, Ser. B, 57 (1993), pp. 122-130.  C. J. H. McDiarmid and A. , 111 (1993), pp. 389-392.  J. C. Meyer, Nombre chromatique total d'un hypergraphe, J. Comb. Theory, Ser. B, 24 (1978), pp. 44-50. 26 M. BEHZAD  C. St. J. A. Nash-Williams, The reconltruction problem, in Selected Topics in the Theory of Graphs, L. W. Beineke and R. J. , Academic Press, 1978.  N. P. ubgmph. and total grap'" with cro"ing number 1, J.
We have not explored the recursive techniques here, and refer the interested reader to [6, 8]. J. COLBOURN Let us close with a few open problems. Constructions from pairwise balanced designs aft"ord a number of dramatic improvements on other recursive constructions. Thus it would be particularly interesting to extend the battery of constructions for pairwise balanced designs with large (and hopefully prime power) block sizes. Greig  has made progress in this direction, and it appears to be a very important direction for future research.