Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs

For a graph H, let σ 2(H ) = min{d (u) + d (v ) | for every edge uv ∈ E (H )}. For r > 0 and k ∈ {2, 3}, let Q0(r, k ) be a set of k-edgeconnected K3-free graphs of order at most r and without spanning closed trails. We show that for given p > 0 and , if H is a k-connected claw-free graph of order n with δ(H ) ≥ 3 and σ 2(H ) ≥ (2n + )/p, and if n is sufficiently large, then either H is Hamiltonian or the Ryjác ̆ek’s closure cl (H ) = L(G) where G is an essentially k-edge-connected K3-free graph that can be contracted to a graph in Q0(5p − 10, k ). As applications, we prove: (i) For k = 2, if δ(H ) ≥ 3 and if σ 2(H ) > 2n−6 3 and n is sufficiently large, then H is Hamiltonian. (ii) For k = 3, if σ 2(H ) > 2n+10 10 and n is sufficiently large, then H is Hamiltonian. These bounds are sharp. Furthermore, since the graphs in Q0(5p − 10, k ) are fixed for given p and can be determined in a constant time, any improvement to (i) or (ii) by increasing the value of p and so enlarging the number of exceptions can be obtained computationally. C © 2017 Wiley Periodicals, Inc. J. Graph Theory 00: 1–20, 2017 Journal of Graph Theory C © 2017 Wiley Periodicals, Inc. 1 2 JOURNAL OF GRAPH THEORY