Hamilton-connected {claw, net}-free graphs, I

This is the first one in a series of two papers, in which we complete the characterization of forbidden generalized nets implying Hamilton-connectedness of a 3-connected claw-free graph. In this paper, we first develop the necessary techniques that allow one to handle the problem, namely: We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and third authors, such that not only the line graph preimage of a closure, but also its core has certain strong structural properties. We prove a special version of the "nine-point-theorem" by Holton et al. that allows one to handle Hamilton-connectedness of "small" {K1,3,Ni,j,k} $\{{K}_{1,3},{N}_{i,j,k}\}$-free graphs (where Ni,j,k ${N}_{i,j,k}$ is the graph obtained by attaching endvertices of three paths of lengths i,j,k $i,j,k$ to a triangle). By a combination of these techniques, as an application, we prove that every 3-connected {K1,3,N1,3,3} $\{{K}_{1,3},{N}_{1,3,3}\}$-free graph is Hamilton-connected. (i)(ii)(iii) The paper is followed by its second part in which we show that every 3-connected {K1,3,X} $\{{K}_{1,3},X\}$-free graph, where X is an element of{N1,1,5,N2,2,3} $X\in \{{N}_{1,1,5},{N}_{2,2,3}\}$, is Hamilton-connected. All the results on Hamilton-connectedness are sharp.