On expansive graphs

This article presents results on expansive graphs that Víctor Neumann-Lara obtained in the late 1970’s and reported mostly without proofs in [8] and [9]. The untimely death of Víctor left us to complete this project, which had started in [7]. We had at hand [8, 9] and the manuscript [10] that, using his old notebooks, Víctor wrote with one of us in 1995. The original material has been thoroughly rewritten and recast in the setting of [7]. This allowed for a clearer and more natural presentation. Some shortcuts were found, and applications added. Our graphs are finite, simple and non-empty. We identify induced subgraphs and vertex sets. The clique graph K(G) of a graph G is the intersection graph of its cliques (maximal complete subgraphs, or just maximal completes). The iterated clique graphs K(G) are defined by K(G) = G and K(G) = K(K(G)). We refer to [12, 4, 13] for the literature on iterated clique graphs. In the study of the dynamics of the clique operator K, two types of K-behaviour stand out: G is clique convergent if K(G) ∼= K(G) for some pair n < m, and G is clique divergent if |V (K(G))| tends to infinity with n (iff this sequence is unbounded). A graph is clique divergent if and only if it is not clique convergent. In this paper we study expansivity, a stronger notion than clique divergence. Expansivity works for coaffine graphs. These are graphs with a special kind of symmetry: a fixed automorphism (a coaffination) that maps each vertex out of its closed neighbourhood, as for instance the antipodal maps of the octahedron and the icosahedron. If A and B are coaffine graphs, adding to their disjoint union all possible edges from A to B we obtain their Zykov sum A + B, which is coaffine with the union of the coaffinations of A and B. We will show that, from the additive viewpoint, the great majority of coaffine graphs are expansive: any Zykov sum of at least 3 coaffine summands is expansive. A coaffine subgraph of a coaffine graph is any subgraph (induced or not) which is invariant under the coaffination. If A is an expansive coaffine subgraph of B, then B is expansive. Thus, a coaffine graph does not need to be a Zykov sum to be expansive: it is enough that it has a complete tripartite coaffine subgraph. Furthermore, a complete bipartite coaffine subgraph will suffice if one of the parts induces a connected subgraph: In fact, if G and H are coaffine and H is connected, then G+H is expansive. Moreover, it is not even necessary to contain Zykov sums in order to be expansive: interesting examples include complements and powers of cycles; indeed, the K-behaviour of these complements and powers is completely characterized in this work. A further interesting consequence of the theory is that, save possibly for one, every connected graph each of whose neighbourhoods is either a square or a pentagon is K-divergent. We also show that every graph is an induced subgraph of some expansive graph.