A characterization of graphs with supereulerian line graphs. Issue 1 (2nd January 2020) - Huang, Yufei; He, Weihua; Huang, Guixian; Lai, Hong-Jian; Song, Sulin
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 ) | ≥ 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 .