A characterization of graphs with supereulerian line graphs

Abstract

The line graph $L(G)$ of a graph G is a simple graph with $E(G)$ being its vertex set, where two vertices are adjacent in $L(G)$ whenever the corresponding edges share a common vertex in G. A graph H is even if every vertex of H has even degree, and a graph is supereulerian if it has a spanning closed trail. We obtain a characterization for a graph G to have a supereulerian line graph $L(G)$, as follows: for a connected graph G with $|E(G)|\ge 3$, the line graph $L(G)$ has a spanning closed trail if and only if G has an even subgraph H (possibly null) such that both G remains connected after deleting all degree 2 vertices not in H, and every degree 2 vertex not in H must be adjacent only to vertices of degree at least 3 in G.

Keywords:

• Eulerian
• supereulerian
• line graph
• collapsible graph
• trail

Acknowledgments

The authors would like to thank the referees for their helpful suggestions to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Hong-Jian Lai  http://orcid.org/0000-0001-7698-2125

Additional information

Funding

This research is supported in part by the National Natural Science Foundation of China [grant numbers 11501139, 11971180, 11601093, 11671296, 11771039, and 11771443], the Guangdong Provincial Natural Science Foundation [grant number 2019A1515012052], and the Key Project at School Level of Guangzhou Civil Aviation College [grant number 18X0429].