Spanning eulerian subgraphs in N²-locally connected claw-free graphs

A graph G is N^m -locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. In this note, we first present a counterexample to the conjecture that every 3-connected, N²-locally connected claw-free graph is hamiltonian and then show that both connected N²-locally connected claw-free graph and connected N³-locally connected claw-free graph with minimum degree at least three have connected even [2, 4]-factors.