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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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