### Abstract

A graph is called supereulerian if it has a spanning closed trail. Let $G$ be a 2-edge-connected graph of order $n$ such that each minimal edge cut $E \subseteq E (G)$ with $|E| \le 3$ satisfies the property that each component of $G-E$ has order at least $(n-2)/5$. We prove that either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree $\delta\ge 4$: If $G$ is a 2-edge-connected graph of order $n$ with $\delta (G)\ge 4$ such that for every edge $xy\in E (G)$ , we have $\max \{d(x),d(y)\} \ge (n-7)/5$, then either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. We show that the condition $\delta(G)\ge 4$ cannot be relaxed.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Publication status | Published - 1999 |

### Publication series

Name | |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1507 |

ISSN (Print) | 0169-2690 |

### Keywords

- EWI-3327
- MSC-05C45
- MSC-05C35
- IR-65695

## Cite this

Broersma, H. J., & Xiong, L. (1999).

*On minimum degree conditions for supereulerian graphs*. Enschede: University of Twente, Department of Applied Mathematics.