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Degree sequence and supereulerian graphs. (English) Zbl 1165.05005
Summary: A sequence \(d=(d_{1},d_{2},\ldots ,d_n)\) is graphic if there is a simple graph \(G\) with degree sequence \(d\), and such a graph \(G\) is called a realization of \(d\). A graphic sequence \(d\) is line-hamiltonian if \(d\) has a realization \(G\) such that \(L(G)\) is hamiltonian, and is supereulerian if \(d\) has a realization \(G\) with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence \(d=(d_{1},d_{2},\ldots ,d_n)\) has a supereulerian realization if and only if \(d_n\geq 2\) and that \(d\) is line-hamiltonian if and only if either \(d_{1}=n - 1\), or \(\sum _{d_i=1}d_i\leq \sum _{d_j\geq 2}(d_j - 2)\).

MSC:
05C07 Vertex degrees
05C45 Eulerian and Hamiltonian graphs
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