Nowhere-zero flows in Cartesian bundles of graphs

A nowhere-zero k-flow on a graph G is an assignment of a direction and a non-zero integer in absolute value smaller than k to each edge of G in such a way that, for each vertex, the sum of incoming values equals the sum of outgoing values. Nowhere-zero flow problems evolved from flowcolouring duality to a theory which has a central role in graph theory. There are many important flow problems which are still open – for example the famous Tutte’s 5-flow conjecture proposed in 1954, see [Jaeger, 1988]. Connections between nowhere-zero flows and other areas of graph theory such as colouring, matching, graph embedding and graph symmetry, provide strong motivation for further study. Our work continues and extends the study of nowhere-zero flows on product graphs initiated by Imrich and Škrekovski [Imrich and Škrekovski, 2003]. Product graphs have been examined for many different graph properties because of their relatively simple structure and considerable generality. The first significant result concerning flows on Cartesian products of graphs is due to Imrich and Škrekovski [Imrich and Škrekovski, 2003]. It states that the Cartesian product of any two nontrivial connected graphs has a nowhere-zero 4-flow. We have examined a natural (although lesser known) generalisation of the Cartesian product called Cartesian bundle. The concept was introduced in 1982 by Pisanski and Vrabec [Pisanski and Vrabec, 1982] and subsequently studied by several authors. Given two graphs, a base graph B and a fibre F , a Cartesian bundle B φF is a graph with vertex-set V (B)×V (F) constructed similarly as the Cartesian product B F except the following: for each edge uv of B, the usual identical isomorphism between the F-layers u F and v F induced by the subgraph uv F – that is, the isomorphism (u,x) 7→ (v,x) – can be replaced by any other isomorphism φuv. The concept of a Cartesian ∗[email protected] †[email protected] bundle thus embodies the idea of a graph that locally resembles the Cartesian product but globally may have a different structure. We have proved the following statement.