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On 3-edge-connected supereulerian graphs. (English) Zbl 1235.05084
Summary: The supereulerian graph problem, raised by F.T. Boesch, C. Suffel, and R. Tindell [“The spanning subgraphs of Eulerian graphs”, J. Graph Theory 1, 79–84 (1977; Zbl 0363.05042)], asks when a graph has a spanning eulerian subgraph. Pulleyblank showed that such a decision problem, even when restricted to planar graphs, is NP-complete. Jaeger and Catlin independently showed that every 4-edge-connected graph has a spanning eulerian subgraph. In 1992, Zhan showed that every 3-edge-connected, essentially 7-edge-connected graph has a spanning eulerian subgraph. It was conjectured in 1995 that every 3-edge-connected, essentially 5-edge-connected graph has a spanning eulerian subgraph. In this paper, we show that if \(G\) is a 3-edge-connected, essentially 4-edge-connected graph and if for every pair of adjacent vertices \(u\) and \(v\), \(d_{G}(u) + d_{G}(v) \geq 9\), then \(G\) has a spanning eulerian subgraph.
MSC:
05C45 Eulerian and Hamiltonian graphs
05C40 Connectivity
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