Neighborhood intersections and Hamiltonicity in almost claw-free graphs (Q5957705)

The partially square \(G^*\) of a graph \(G\) is a graph obtained from \(G\) by adding edges \(uv\) satisfying the conditions \(uv\not\in E(G)\), and there is some \(w\in N(u)\cap N(v)\) such that \(N(w)\subseteq N(u)\cup N(v)\cup\{u,v\}\). Let \(d(x)\) be the degree and \(d(x,A)\) be the distance between the vertex \(x\) and the set \(A\subseteq V(G)\). For an integer \(p\geq 2\) and a set \(Y\subseteq V(G)\), let \(n(Y)=|\{v\in V(G):d(v,Y)\leq 2\}|\) and \(I_p(G)=\{U:U\subseteq V(G)\) independent with \(|U|=p\}\). The author proves that a \(k\)-connected, almost claw-free graph \(G\) with \(k\geq 2\) is Hamiltonian if \(\sum_{u\in U}d(u)\geq n(U)-k\) for each \(U\in I_{k+1}(G^*)\). In particular, this result confirms a 1996 conjecture proposed by \textit{H. J. Broersma, Z. Ryjáček}, and \textit{I. Schiermeyer} [J. Graph Theory 21, No. 4, 431-439 (1996; Zbl 0847.05069)].