A condition for the three colourability of planar locally path graphs
A graph G is said to be locally path if for every vertex the subgraph induced by its neighbours is a path. Planar locally path graphs are a natural generalization of maximal outerplanar graphs. We show that they have a recursive construction which generalizes that of maximal outerplanar graphs. Using this characterization we obatin a ’local’ condition for the three colourability of planar locally path graphs. As a corollary, we show that maximal planar graphs are three colourable iff every vertex has even degree. We also derive a structural property of maximal planar graphs.
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