Contractible graphs for flow index less than three

Abstract

For a graph G, the flow indexϕ(G) is the smallest rational number t>0 such that the graph has a circular t-flow. Li et al. (2018) recently proved that ϕ(G)<3 for any 8-edge-connected graph G, and conjectured that 6-edge-connectivity would suffice. Here we present a contraction method to investigate this problem and apply it to verify this conjecture for certain 6-edge-connected graph families, including chordal graphs, graphs with few odd vertices, and graphs with small independence number.