Edge Degree Conditions for 2-Iterated Line Graphs to Be Traceable

The line graph L(G) of G has E(G) as its vertex set, and two vertices are adjacent in L(G) if and only if the corresponding edges share a common end vertex in G. Let σ¯2(G)=min{dG(x)+dG(y)|xy∈E(G)}. We show that, if σ¯2(G)≥2(⌊n11⌋−1) and n is sufficiently large, then either L(L(G)) is traceable or the Veldman’s reduction G′ is one of well-defined classes of exceptional graphs. Furthermore, if σ¯2(G)≥2(⌊n7⌋−1) and n is sufficiently large, then L(L(G)) is traceable. The bound 2(⌊n7⌋−1) is sharp. As a byproduct, we characterize the structure of a connected graph with a non-traceable 2-iterated line graph.