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We estimate the number of possible degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs
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