Recharacterizing Eulerian: Intimations of new duality.

*(English)*Zbl 0547.05043A new characterization of Eulerian graphs is presented. It is formulated generally in terms of binary matroids. The edge-disjoint unions of circuits (or of cutsets) are called circs (or segs respectively). The set of all circs in a given binary matroid is denoted by \({\mathcal C}^+\), the set of all segs by \({\mathcal D}^+\). The symbol \(R_ e\), where e is an edge, denotes the set consisting of e and all edges x with the property that the number of circuits which contain both e and x is odd. Theorem. The following are equivalent for all binary matroids: (i) Each \(R_ e\) meets each seg evenly. (ii) Each \(R_ e\) is a circ. (iii) Every edge is contained in an odd number of circuits. (iv) Every cutset contains an even number of edges. This theorem has four corollaries; we quote three of them. Corollar 1. A connected graph is Eulerian if and only if each edge is in an odd number of circuits. Corollary 2. A connected graph is Eulerian if and only if for every edge e the set \(R_ e\) of edges induces a subgraph with Eulerian components. Corollary 3. A graph is bipartite if and only if every edge is in an odd number of cutsets.

Reviewer: B.Zelinka

##### MSC:

05C45 | Eulerian and Hamiltonian graphs |

05C75 | Structural characterization of families of graphs |

05B35 | Combinatorial aspects of matroids and geometric lattices |

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##### References:

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