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) ≥ ∆+ (G), where ∆+ (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 fH 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 fH a...

• Type: Journal
• Year: 2011
• Where: DAM
• Authors: Christine T. Cheng, Ichiro Suzuki
• Keywords: Coding Function, DAM 2011, Smallest Integer, Theoretical Computer Science, Weak Sense