Traceability in {K _1,4, K _1,4 + e}-free graphs

A graph G is called {H₁, H₂,…, H_k}-free if G contains no induced subgraph isomorphic to any graph H_i, 1 ⩽ i ⩽ k. We define σk=min{∑i=1kd(vi):{v1,…,vk}is an independent set of vertices inG}. In this paper, we prove that (1) if G is a connected {K_1,4, K_1,4 + e}-free graph of order n and σ₃(G) ⩾ n − 1, then G is traceable, (2) if G is a 2-connected {K_1,4, K_1,4 + e}-free graph of order n and ∣N(x₁) ∪ N(x₂)∣ + ∣N(y₁) ∪ N(y₂)∣ ⩾ n − 1 for any two distinct pairs of non-adjacent vertices {x₁, x₂}, {y₁, y₂} of G, then G is traceable, i.e., G has a Hamilton path, where K_1,4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K_1,4.