On (k,d)-colorings and fractional nowhere-zero flows

Abstract

The concepts of (k, d)-coloring and the star chromatic number, studied by Vince, by Bondy and Hell, and by Zhu are shown to reflect the cographic instance of a wider concept, that of fractional nowhere-zero flows in regular matroids.

Eulerian and Bipartite Orientable Matroids

“…Applying duality to planar graphs, these notions produce characterizations of bipartite graphs. Indeed the notions of flows and colourings of regular matroids can be formulated in terms of orientations, as was observed by Goddyn et al [2]. The equivalent connection for graphs had been made by Minty [3].…”

The circular chromatic number of hypergraphs

“…The same results, often in the context of uniform hypergraphs and derived using a slightly different definition of homomorphism, have been noted by Eslahchi and Rafiey [6]. After describing the framework, it is shown how four other formulations of the circular chromatic number of graphs, due respectively to Vince [14], Zhu [18], Barbosa and Gafni [2] (also see the paper by Yeh and Zhu [16]), and Goddyn, Tarsi and Zhang [8] can be extended to hypergraphs. The modifications of the definitions involved are in the same spirit in each case, and the statements and proofs are substantially the same as for graphs.…”

“…We explore the relationship between the imbalance and the circular chromatic number of a hypergraph and deduce a classical result of Minty [11] regarding the chromatic number. The work in this section closely follows that of Brewster and Hell [4], who gave the first graph theoretic proof of a theorem found by Goddyn, Tarsi and Zhang [8].…”

Flows, flow-pair covers and cycle double covers

“…From all those approaches, one might propose that the famous 5-flow conjecture by Tutte [16] could be approached by covering a bridgeless graph with a pair of a 2-flow and a 5 2 -flow ( 5 2 -flow is defined as a circular flow, see [6,20] for definition). However, this approach does not work for cubic graphs since the support of a 5 2 -flow must be odd-5-edge-connected.…”