THE SEPARATED BOX PRODUCT OF TWO DIGRAPHS PRIMO ˇ Z POTO ˇ CNIK AND STEVE WILSON Abstract. A new product construction of graphs and digraphs, based on the standard box product of graphs and called the separated box product, is pre- sented, and several of its properties are discussed. Questions about the sym- metries of the product and their relations to symmetries of the factor graphs are considered. An application of this construction to the case of tetravalent edge-transitive graphs is discussed in detail. 1. Introduction A dart (also known as an arc) of a graph Γ is an ordered pair of adjacent vertices of Γ, and the set of all darts of Γ is denoted by D(Γ). A graph Γ is dart-transitive provided that its symmetry group Aut(Γ) acts transitively on D(Γ). It was proved in [13] that apart from a well-understood infinite family of graphs and finitely many exceptions, the order of the vertex-stabiliser G v in the symmetry group G of a connected tetravalent dart-transitive graph of order n satisfies the inequality n ≥ 2|G v | log 2 (|G v |/2). The largest amongst the finite set of exceptions is a graph (let’s denote it by Γ 8100 ) of order 8100 (appearing in the last line of [13, Table 2]) whose symmetry group is isomorphic to the group PΓL(2, 9) wr C 2 , and has the vertex-stabiliser of large order, 512. This graph was originally constructed as a coset graph of the group PΓL(2, 9) wr C 2 , but an interesting combinatorial construction of Γ 8100 was also given in [13, Section 2.2]: For a graph Λ, let A 2 G(Λ), the ”squared-arc graph” of Λ, be the graph with vertex-set being D(Λ) ×D(Λ) and with two vertices (x,y), (w,z) ∈D(Λ) ×D(Λ) adjacent in A 2 G(Λ) if and only if y = w, x =(v 1 ,v 2 ) and z =(v 2 ,v 3 ) for some vertices v 1 ,v 2 ,v 3 of Λ such that v 1 = v 3 . The graph Γ 8100 is then isomorphic to the graph A 2 G(Λ), where Λ is the Tutte 8-cage (the unique dart-transitive 3-regular graph on 30 vertices with girth 8). It is not at all obvious in what way (if at all) the symmetry group of the Tutte’s 8- cage Λ, isomorphic to PΓL(2, 9), yields the group PΓL(2, 9) wr C 2 acting on A 2 G(Λ). Our endeavour to understand the relationship between Aut(Λ) and Aut(A 2 G(Λ)) led us to a discovery of a surprisingly simple product operation on digraphs, called the separated box product, which significantly generalises the A 2 G construction on one hand and explains many symmetries that A 2 G(Λ) possesses on the other hand. It is the aim of this paper to present the separated box product construction and some of its properties. The construction is described in Section 3.1. In Section 3.2, the symmetry properties of the resulting (di)graph are discussed, and in Section 3.3 the question of its connectedness of is addressed. The relationship between the A 2 G 2000 Mathematics Subject Classification. 20B25. Key words and phrases. digraph, graph, transitive, product. Supported in part by the Slovenian Research Agency, projects J1-5433, J1-6720, and P1-0294. 1 arXiv:1511.00623v1 [math.CO] 2 Nov 2015