Recent examples in the theory of partition graphs

DeTemple, D.W.. M.J. Dineen, J.M. Robertson and K.L. McAvaney, theory of partition graphs, Discrete Mathematics 113 (1993) 255-258. examples in the A partition graph is an intersection graph for a collection of subsets of a universal set S with the property that every maximal independent set of vertices corresponds to a partition of S. Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided. I. The resolution of two questions in partition graph theory A graph G is a general partition graph [2] if there is some set S and an assignment of subsets St, c S to the vertices 21 E V(G) such that: (i) uu E E(G) if and only if St, fl S*, # 0; (ii) S = UuEvcGl S,;and (iii) every maximal independent set of vertices M partitions S into a disjoint collection {S,,l: m E M}. If, furthermore, (iv) u f+S,, #S,, then G is a partition graph [ 1, 21. Correspondence to: D.W. DeTemple, Washington State University, Pullman, WA 99164-3113, USA. 0012-365X/93/$06.00 @ 1993 Elsevier Science Publishers B.V. All rights reserved 256 D. W. DeTemple et al. It has been shown in [2] that St, = S,, in a general partition graph precisely when the vertices u and v have the same closed neighborhoods, so that N[u] = N[v]. Moreover if a graph G has two vertices u and v for which N[u] = N[v], then G is a general partition graph if and only if G u is a general partition graph. Thus to ascertain if a graph is a general partition graph it suffices to remove, successively, all vertices which have the same closed neighborhood as another vertex, and then to check if the reduced graph (with distinct closed neighborhoods) is a partition graph. In this note we answer two questions which arose in the study of partition graphs. The questions pertain to the conditions below on a graph G. In Condition I a clique cover of G is a collection of cliques for which every vertex and edge of G is a member of some clique in the collection. Triangle Condition T. For every maximal independent set M of vertices in G and every edge WJ in G M, there is a vertex m E M such that uvm is a triangle in G. Incidence Condition I. There is a clique cover F of G with the property that every maximal independent set has a vertex from each clique in r. Condition T is known [l, 21 to be necessary for a graph to be a general partition graph. Condition I is both necessary and sufficient [2]. Condition T has been useful screening criterion to check if a given graph might be a partition graph. Indeed, since no contrary examples were known, it had been asked ([ 1,2,4]): Is Condition T also a sufficient condition? A computer search of all graphs through ten vertices has shown that Condition T is not sufficient. The smallest example occurs on nine vertices, and is the graph Gr shown in Fig. 1. There are five maximal independent sets of vertices, { vl, v3, us}, {q,, v4, v,}, (v,, v4, v7}, {I+, v5, v,} and {v3, v6, v,}, and it is straightforward to verify that Condition T holds for GP Fig. 1. A graph satisfying Condition T which is not a general partition graph.