On extremal \(k\)-supereulerian graphs. (English) Zbl 1277.05104
Summary: A graph \(G\) is called \(k\)-supereulerian if it has a spanning even subgraph with at most \(k\) components.
In this paper, we prove that any 2-edge-connected loopless graph of order \(n\) is \(\lceil (n-2)/3\rceil\)-supereulerian, with only one exception. This result solves a conjecture in [Z. Niu and L. Xiong, Australas. J. Comb. 48, 269–279 (2010; Zbl 1232.05185)].
As applications, we give a best possible size lower bound for a 2-edge-connected simple graph \(G\) with \(n>5k+2\) vertices to be \(k\)-supereulerian, a best possible minimum degree lower bound for a 2-edge-connected simple graph \(G\) such that its line graph \(L(G)\) has a 2-factor with at most \(k\) components, for any given integer \(k>0\), and a sufficient condition for \(k\)-supereulerian graphs.
In this paper, we prove that any 2-edge-connected loopless graph of order \(n\) is \(\lceil (n-2)/3\rceil\)-supereulerian, with only one exception. This result solves a conjecture in [Z. Niu and L. Xiong, Australas. J. Comb. 48, 269–279 (2010; Zbl 1232.05185)].
As applications, we give a best possible size lower bound for a 2-edge-connected simple graph \(G\) with \(n>5k+2\) vertices to be \(k\)-supereulerian, a best possible minimum degree lower bound for a 2-edge-connected simple graph \(G\) such that its line graph \(L(G)\) has a 2-factor with at most \(k\) components, for any given integer \(k>0\), and a sufficient condition for \(k\)-supereulerian graphs.
MSC:
| 05C45 | Eulerian and Hamiltonian graphs |
Citations:
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