Circular flows via extended Tutte orientations

Abstract

This paper consists of two major parts. In the first part, the relations between Tutte orientations and circular flows are explored. Tutte orientation (modulo orientation) was first observed by Tutte for the study of 3-flow problem, and later extended by Jaeger for circular (2+1/p)-flows. In this paper, it is extended for circular λ-flows for all rational numbers λ. This theorem is one of the key tools in the second part of the paper. It was proved by Lovász et al. (2013) [8] that every (6p+1)-odd-edge-connected graph admits a circular (2+1/p)-flow. In the second part, this result is further extended to other odd integers 6p+3 and 6p−1 for any positive integer p. We show that (i) every(6p−1)-odd-edge-connected graph admits a circular(2+22p−1)-flow, and (ii) every(6p+3)-odd-edge-connected graph has flow index strictly less than2+1/p. Both (i) and (ii) generalize some early results. For example, the case p=1 of (i) is the well known 4-flow theorem of Jaeger (1979) [4], and the case p=1 of (ii) is a recent result by Thomassen and the authors (2018) [7]. The proofs of (i) and (ii) provide (principally different) new approach to those previous results.