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, \(N^2\)-locally connected claw-free graph is hamiltonian and then show that both connected \(N^2\)-locally connected claw-free graph and connected \(N^3\)-locally connected claw-free graph with minimum degree at least three have connected even \([2, 4]\)-factors.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.