Cereceda's conjecture

Unsolved problem in the mathematics of graph coloring

In the mathematics of graph coloring, Cereceda’s conjecture is an unsolved problem on the distance between pairs of colorings of sparse graphs. It states that, for two different colorings of a graph of degeneracy d, both using at most d + 2 colors, it should be possible to reconfigure one coloring into the other by changing the color of one vertex at a time, using a number of steps that is quadratic in the size of the graph. The conjecture is named after Luis Cereceda, who formulated it in his 2007 doctoral dissertation.

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