Tiling edge-coloured graphs with few monochromatic bounded-degree graphs

Abstract

We prove that for all integers $$\Delta ,r \ge 2$$ , there is a constant $$C = C(\Delta ,r) >0$$ such that the following is true for every sequence $${\mathcal {F}}= \{F_1, F_2, \ldots \}$$ of graphs with $$v(F_n) = n$$ and $$\Delta (F_n) \le \Delta $$ , for each $$n \in {\mathbb {N}}$$ . In every r-edge-coloured $$K_n$$ , there is a collection of at most C monochromatic copies from $${\mathcal {F}}$$ whose vertex-sets partition $$V(K_n)$$ . This makes progress on a conjecture of Grinshpun and Sárközy.