The clique operator on matching and chessboard graphs

Abstract

Given positive integers m, n, we consider the graphs G n and G m,n whose simplicial complexes of complete subgraphs are the well-known matching complex M n and chessboard complex M m,n . Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy equivalent to their clique graphs. We determine the clique behavior of the chessboard graph G m,n in terms of m and n, and show that G m,n is clique-divergent if and only if it is not clique-Helly. We give partial results for the clique behavior of the matching graph G n .

On the clique behavior of circulants with three small jumps

“…It is known that every Helly graph is clique convergent [7], but the opposite is not true in general [7,4,13,9]. However, there are several families of graphs which are known to be divergent precisely when they are not Helly such as cographs [10], complements of cycles [20,15], powers of cycles [15] and chessboard graphs [16]. We shall prove that the circulants studied here also exhibit this property.…”

On the homotopy type of the iterated clique graphs of low degree

“…Theorem 2.4. (Theorem 4.2 from [14]) Let G be an O 3 -free graph such that every triangle in G is contained in a unique clique. Then G is homotopy Kinvariant.…”

On the clique behavior and Hellyness of the complements of regular graphs

“…However, there are several graph classes for which it is has been proven that a graph in such a class is convergent if and only if it is Helly. Such classes include that of cographs ( [9]), complements and powers of cycles ( [15], [10]), chessboard graphs ( [12]) and circulants with three small jumps ( [11]).…”