Eulerian Subgraphs in a Class of Graphs

Abstract

Let \(G\) be a graph and let \(D_1(G)\) denote the set of vertices of degree one in \(G\). In [1], Behocine, Clark, Kéhler, and Veldman conjectured that for a connected simple graph \(G\) of \(n\) vertices, if \(G – D_1(G)\) is \(2\)-edge-connected, and if for any edge \(xy \in E(G)\), \(d(x) + d(y) > \frac{2n}{5}-2\), then \(L(G)\) is hamiltonian.

In this note, we shall show that the conjecture above holds for a class of graphs that includes the \(K_{1,3}\)-free graphs, and we shall also characterize the extremal graphs.