Factorial properties of graphs

We explore an operation on graphs that is analogous to the factorial of a positive integer. We develop formulas for the factorials of complete graphs, odd cycles and complete bipartite graphs, and we prove some properties of the factorial operation. We conclude with a list of open questions. This paper explores some properties of a factorial operation that associates to each graph G a graph G! which we call the factorial of G. In [1] this operation is used to derive some cancellation properties of the direct product of graphs. However, our purpose here is simply to investigate the factorial operation on several classes of graphs and to deduce some of its properties. Before stating our main definition, we set up notation and recall some relevant constructions. All our graphs are finite, undirected and are allowed to have loops. We thus regard a graphG = (V (G), E(G)) as a symmetric relationE(G) on the set V (G). The complete graph on n vertices is denoted Kn, while K ∗ n denotes the complete graph with loops at each vertex. We put V (Kn) = V (K ∗ n) = {0, 1, 2, 3, . . . , n − 1}. The complete bipartite graph with partite sets of sizes m and n is denoted K(m,n). The complement of a graph G is denoted G. The direct product of two graphs G and H is the graph G×H whose vertex set is the Cartesian product V (G)× V (H) and whose edges are the pairs (g, h)(g′, h′) with gg′ ∈ E(G) and hh′ ∈ E(H). (See [2] for a standard reference for the direct product.) We denote by G+H the disjoint union of G and H. Given a positive integer n, we let nG denote the graph consisting of n copies of G. An involution is an automorphism of G that is its own inverse. 1 The Graph Factorial The following definition, introduced in [1], is the main topic of this paper. 266 G. ABAY-ASMEROM, R.H. HAMMACK AND D.T. TAYLOR Definition 1 The factorial of a graph G is the graph, denoted G!, whose vertices are the permutations of V (G). Permutations λ and μ are adjacent in G! exactly when gg′ ∈ E(G) ⇐⇒ λ(g)μ−1(g′) ∈ E(G) for all pairs g, g′ ∈ V (G). We denote an edge joining vertices λ and μ as (λ)(μ) in order to avoid confusion with composition. We remark that E(G!) is indeed a symmetric relation on V (G!) because replacing g and g′ in the definition with λ−1(g) and μ(g′) gives λ−1(g)μ(g′) ∈ E(G) ⇐⇒ gg′ ∈ E(G), from which symmetry of G yields gg′ ∈ E(G) ⇐⇒ μ(g)λ−1(g′) ∈ E(G). Observe that if λ is an automorphism of G, then (λ)(λ−1) ∈ E(G!), but it is not necessarily true that every edge of G! has this form. Notice also that there is a loop at a vertex α of G! precisely when α satisfies gg′ ∈ E(A) ⇐⇒ α(g)α−1(g′) ∈ E(G) for all pairs g, g′ ∈ V (G). Such an α is called an anti-automorphism in [1], and it is proved there that for any bipartite graph B, the condition G× B ∼= H × B implies G ∼= H if and only if every anti-automorphism α of G factors as α = λμ for some (λ)(μ) ∈ E(G!). We are not concerned with such applications here. Our purpose is merely to determine factorials of some elementary graphs.