ANTI-RAMSEY NUMBER OF EDGE-DISJOINT RAINBOW SPANNING TREES IN ALL GRAPHS.

An edge-colored graph G is called rainbow if every edge of G receives a different color. Given any host graph G, the anti-Ramsey number of t edge-disjoint rainbow spanning trees in G, denoted by r(G,t), is defined as the maximum number of colors in an edge-coloring of G containing no t edge-disjoint rainbow spanning trees. For any vertex partition P, let E(P,G) be the set of non-crossing edges in G with respect to P. In this paper, we determine r(G,t) for all host graphs G: r(G,t)=|E(G)| if there exists a partition P0 with |E(G)|-|E(P0,G)|<t(|P0|-1); and r(G,t)=maxP:|P|=3{|E(P,G)|+t(|P|-2)} otherwise. As a corollary, we determine r(Kp,q ,t) for all values of p,q,t, improving a result of Jia, Lu and Zhang. [ABSTRACT FROM AUTHOR]