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Let G be an n-vertex graph that contains linearly many cherries (i.e., pathson 3 vertices), and let c be a coloring of the edges of the complete graph K_nsuch that at each vertex every color appears only constantly many times. In1979, Shearer conjectured that such a coloring c must contain a properlycolored copy of G. We establish this conjecture in a strong form, showing thatit holds even for graphs G with O(n^(4/3)) cherries and moreover this bound onthe number of cherries is best possible up to a constant factor. We also provethat one can find a rainbow copy of such G in every edge-coloring of K_n inwhich all colors appear bounded number of times.  Our proofs combine a framework of Lu and Szekely for using the lopsidedLovasz local lemma in the space of random bijections together with someadditional ideas.

Properly colored and rainbow copies of graphs with few cherries