Degree sums of adjacent vertices for traceability of claw-free graphs

Abstract

The line graph of a graph G, denoted by L(G), has E(G) as its vertex set, where two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common. For a graph H, define ${\overline \sigma _2}(H) = \min \left\{{d(u) + d(v):\;\;uv\; \in \;E(H)} \right\}$. Let H be a 2-connected claw-free simple graph of order n with δ(H) ⩾ 3. We show that, if ${\overline \sigma _2}(H)\geqslant {1 \over 7}(2n - 5)$ and n is sufficiently large, then either H is traceable or the Ryjáček’s closure cl(H) = L(G), where G is an essentially 2-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have no spanning trail. Furthermore, if ${\overline \sigma _2}(H)\; > \;{1 \over 3}(n - 6)$ and n is sufficiently large, then H is traceable. The bound ${1 \over 3}(n - 6)$ is sharp. As a byproduct, we prove that there are exactly eight graphs in the family ${\cal G}$ of 2-edge-connected simple graphs of order at most 11 that have no spanning trail, an improvement of the result in Z. Niu et al. (2012).