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.

Abstract. A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.

Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A ⊆ V . An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P . (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Mader's S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k − 2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem.

The List Edge Colouring Conjecture asserts that, given any multigraph G with chromatic index k and any set system {Se : e C E(G)} with each [Se[ = k, we can choose elements Se E Se such that Ser whenever e and f are adjacent edges. Using a technique of Alon and Tarsi which involves the graph monomial ]-I{xu -Xv : uv E E} of an oriented graph, we verify this conjecture for certain families of 1-factorable multigraphs, including 1-factorable planar graphs.

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