an:06566334
Zbl 1333.05226
Larri??n, F.; Piza??a, M. A.; Villarroel-Flores, R.
On self-clique shoal graphs
EN
Discrete Appl. Math. 205, 86-100 (2016).
00354433
2016
j
05C69
clique graphs; self-clique graphs; constant link
Summary: The clique graph of a graph \(G\) is the intersection graph \(K(G)\) of its (maximal) cliques, and \(G\) is self-clique if \(K(G)\) is isomorphic to \(G\). A graph \(G\) is locally \(H\) if the neighborhood of each vertex is isomorphic to \(H\). Assuming that each clique of the regular and self-clique graph \(G\) is a triangle, it is known that \(G\) can only be \(r\)-regular for \(r \in \{4, 5, 6 \}\) and \(G\) must be, depending on \(r\), a locally \(H\) graph for some \(H \in \{P_4, P_2 \cup P_3, 3 P_2 \}\). The self-clique locally \(P_4\) graphs are easy to classify, but only a family of locally \(H\) self-clique graphs was known for \(H = P_2 \cup P_3\), and another one for \(H = 3 P_2\).
We study locally \(P_2 \cup P_3\) graphs (i.e. shoal graphs). We show that all previously known shoal graphs were self-clique. We give a bijection from (finite) shoal graphs to 2-regular digraphs without directed 3-cycles. Under this translation, self-clique graphs correspond to self-dual digraphs, which simplifies constructions, calculations and proofs. We compute the numbers, for each \(n \leq 28\), of self-clique and non-self-clique shoal graphs of order \(n\), and also prove that these numbers grow at least exponentially with \(n\).