Graphs with the circuit cover property

Alspach, Brian; Goddyn, Luis; Zhang, Cun Quan

doi:10.1090/S0002-9947-1994-1181180-1

Graphs with the circuit cover property
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by Brian Alspach, Luis Goddyn and Cun Quan Zhang PDF

Trans. Amer. Math. Soc. 344 (1994), 131-154 Request permission

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.

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On compatible cycle decompositions of eulerian graphs

On even cycle decompositions of eulerian graphs