Weak sense of direction labelings and graph embeddings

An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences α and α ′ generated by non-trivial walks on G starting at u , f ( α ) = f ( α ′ ) if and only if the two walks end at the same node. The function f is referred to as a coding function of λ . The weak sense of direction number of G , WSD ( G ) , is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD ( G ) ≥ Δ + ( G ) , where Δ + ( G ) is the maximum outdegree of G . Let us say that a function τ : V ( G ) → V ( H ) is an embedding from G onto H if τ demonstrates that G is isomorphic to a subgraph of H . We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f H as a coding function if and only if G can be embedded onto H . Additionally, we show that the problem “Given G , does G have a WSD-labeling that uses a particular coding function f ?” is NP-complete even when G and f are fairly simple. Second, when D is a distributive lattice, H ( D ) is its Hasse diagram and G ( D ) is its cover graph, then WSD ( H ( D ) ) = Δ + ( H ( D ) ) = d ∗ , where d ∗ is the smallest integer d so that H ( D ) can be embedded onto the d -dimensional mesh. Along the way, we also prove that the isometric dimension of G ( D ) is its diameter, and the lattice dimension of G ( D ) is Δ + ( H ( D ) ) . Our WSD-labelings are poset-based, making use of Birkhoff’s characterization of distributive lattices and Dilworth’s theorem for posets.