A characterization of graphs with supereulerian line graphs
Yufei Huang, Weihua He, Guixian Huang, Hong‐Jian Lai, Sulin Song · International Journal of Computer Mathematics Computer Systems Theory · 2019
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)|≥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.