Cliques in exact distance powers of graphs of given maximum degree

The exact distance p-power of a graph G, denoted G[#p], is a graph on vertex set V(G) in which two vertices are adjacent if they are at distance exactly p in G. Given integers k and p, we define f(k, p) to be the maximum possible order of a clique in the exact distance p-powers of graphs with maximum degree k + 1. It is easily observed that f(k, 2) ≤ k2+ k + 1. We prove that equality may only hold if a connected component of G is isomorphic to a member of the class Pk of incidence graphs of finite projective k-geometries. (These famous combinatorial structures are known to exist when k is a prime power, and are conjectured not to exist for other values of k.) We then study the case of graphs of maximum degree k + 1 with clique number k2+ k. One way to obtain such a graph is to remove a vertex from a graph in P k; we call Pk' the class of all such resulting graphs. We prove that for any graph G of maximum degree k + 1 whose exact square has a (k2+ k)-clique, either G has a subgraph isomorphic to a graph in P'k, or a connected component of G is a (k + 1)-regular bipartite graph of order 2(k2+ k). We call Okthe class of such bipartite graphs, and study their structural properties. These properties imply that (if they exist) the graphs in Okmust be highly symmetric. Using this structural information, we show that O2contains only one graph, known as the Franklin graph. We then show that O3also consists of a single graph, which we build. Furthermore, we show that O4and O5are empty. For general values of p, we prove that f(k, p) ≤ (k + 1)k[p/2]+ 1, and that the bound is tight for every odd integer p ≥ 3. This implies that f(k, 2) = f(k, 3) whenever there exists a finite projective k-geometry, however, in such a case, the bound of f(k, 3) could also be reached by highly symmetric graphs built from a finite k-geometry, which is not the case for other values of k.