A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs

We give a simpler proof of the well-known result of Matthews and Sumner stating that squares of connected claw-free graphs are vertex-pancyclic. Contrary to the previous proof, our approach does not resort to Fleischner’s result stating that, when restricted to squares of graphs, vertex-pancyclicity and Hamiltonicity are equivalent. The same proof idea already yielded that connected claw-free graphs of even order have a perfect matching, which is another result of Sumner. We conclude by observing that this proof identifies a larger collection of graphs for which the two properties in question hold.