Cycle lengths and circuit matroids of graphs
For a graph \(G\) with edges \(e_ 1\), \(e_ 2,\dots,e_ m\) choose one of the two orientations of \(e_ i\) \((i=1,2,\dots,m)\) at random. The chain group \(C(\vec G)\) of the oriented graph \(\vec G\) is the free abelian group with the oriented edges \(\vec e_ 1\), \(\vec e_ 2,\dots,\vec e_ m\) as generators; its elements are \(c=\sum c_ ie_ i\) \((c_ 1,\dots,c_ m\) are integers, \(\sum^ m_{i=1}| c_ i|\) is the length of \(c)\). For each circuit \(C\) of \(G\), choose one of the two orientations of \(C\) at random so that it defines a circuit chain. Let \(Z(\vec G)\) be the subgroup of \(C(\vec G)\) generated by the circuit chains. \(Z(\vec G)\) is called the cycle group of \(G\). The purpose of the paper is to prove the following result: Let \(G\) and \(H\) be graphs without cut-edges. Then there is a length preserving group-isomorphism between the cycle groups of \(G\) and \(H\) iff \(G\) and \(H\) have isomorphic circuit matroids.
scientific article; zbMATH DE number 94364